1 North Seattle College Intermediate Algebra Student Workbook Development Team (Scottsdale C.C.) Donna Gaudet William Meacham Jenifer Bohart Amy Volpe...

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Intermediate Algebra Student Workbook

Development Team (Scottsdale C.C.) Donna Gaudet William Meacham Jenifer Bohart Amy Volpe Linda Knop Donna Guhse

Fourth Edition 2014

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This work is licensed (CC-BY) under a Creative Commons Attribution 4.0 International License

About this Workbook Mathematics instructors at Scottsdale Community College in Scottsdale, Arizona created this workbook. The included content is designed to lead students through Intermediate Algebra, from a functions modeling approach, and to develop a deep understanding of the concepts associated with functions, data and change. The included curriculum is broken into twelve lessons. Each lesson includes the following components: MINI-LESSON

The Mini-Lesson is the main instructional component for each lesson. Ideas are introduced with practical applications. Worked Examples are provided for each topic in the Mini-Lesson. Read through these examples carefully. Use these as a guide for completing similar problems. Media Examples can be worked by watching online videos and taking notes/writing down the problem as written by the instructor. Video links can be found at http://sccmath.wordpress.com or may be located within the MathAS Online Homework Assessment System. You-Try problems help reinforce Lesson concepts and should be worked in the order they appear showing as much work as possible. Answers can be checked in Appendix A.

PRACTICE PROBLEMS

This section follows the Mini-Lesson. If you are working through this material on your own, the recommendation is to work all practice problems. If you are using this material as part of a formal class, your instructor will provide guidance on which problems to complete. Your instructor will also provide information on accessing answers/solutions for these problems.

LESSON ASSESSMENT

The last part of each Lesson is a short assessment. If you are working through this material on your own, use these assessments to test your understanding of the lesson concepts. Take the assessments without the use of the book or your notes and then check your answers. If you are using this material as part of a formal class, your instructor will provide instructions for completing these problems and for obtaining solutions to the practice problems.

MATHAS ONLINE HOMEWORK ASSESSMENT SYSTEM

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If you are using these materials as part of a formal class and your class utilizes an online homework/assessment system, your instructor will provide information as to how to access and use that system in conjunction with this workbook.

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Table of Contents Lesson 1 – Introduction to Functions.............................................................................................. 8 Mini-Lesson 1 ........................................................................................................................... 10 Section 1.1 – What is a Function? ........................................................................................ 10 Section 1.2 – Multiple Representations of Functions ........................................................... 14 Section 1.3 – Function Notation ........................................................................................... 18 Section 1.4 – Domain and Range .......................................................................................... 24 Section 1.5 – Applications of Functions ............................................................................... 26 Lesson 1 Practice Problems ...................................................................................................... 32 Lesson 1 Assessment ................................................................................................................ 56 Lesson 2 – Functions and Function Operations ............................................................................ 58 Mini-Lesson 2 ........................................................................................................................... 60 Section 2.1 – Combining Functions ...................................................................................... 60 Section 2.2 – Applications of Function Operations .............................................................. 68 Lesson 2 Practice Problems ...................................................................................................... 70 Lesson 2 Assessment ................................................................................................................ 78 Lesson 3 – Linear Equations and Functions ................................................................................. 79 Mini-Lesson 3 ........................................................................................................................... 81 Section 3.1 – Linear Equations and Functions...................................................................... 81 Section 3.2 – Graphs of Linear Functions ............................................................................ 87 Section 3.3 – Horizontal and Vertical Lines ......................................................................... 90 Section 3.4 – Writing the Equation of a Line ....................................................................... 92 Lesson 3 Practice Problems .................................................................................................... 101 Lesson 3 Assessment .............................................................................................................. 117 Lesson 4 – Linear Functions and Applications ........................................................................... 119 Mini-Lesson 4 ......................................................................................................................... 121 Section 4.1 – Review of Linear Functions .......................................................................... 121 Section 4.2 – Average Rate of Change ............................................................................... 123 Section 4.3 – Scatterplots on the Graphing Calculator ....................................................... 129 Section 4.4 –Linear Regression .......................................................................................... 131 Section 4.5 – Multiple Ways to Determine the Equation of a Line .................................... 136 Lesson 4 Practice Problems .................................................................................................... 137 Lesson 4 Assessment .............................................................................................................. 157 Lesson 5 – Introduction to Quadratic Functions ......................................................................... 159

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Mini-Lesson 5 ......................................................................................................................... 161 Section 5.1 – Characteristics of Quadratic Functions ......................................................... 161 Section 8.2 – Solving Quadratic Equations Graphically .................................................... 169 Section 8.3 – Applications of Quadratic Functions ............................................................ 171 Section 8.4 – Quadratic Modeling ...................................................................................... 175 Lesson 5 Practice Problems .................................................................................................... 177 Lesson 5 Assessment .............................................................................................................. 192 Lesson 6 – Solving Quadratic Equations .................................................................................... 194 Mini-Lesson 6 ......................................................................................................................... 196 Section 6.1 – Quadratic Equations in Standard Form ......................................................... 196 Section 9.2 –Factoring Quadratic Expressions ................................................................... 198 Section 9.3 – Solving Quadratic Equations by Factoring ................................................... 203 Section 9.4 –The Quadratic Formula .................................................................................. 207 Section 9.5 – Complex Numbers ........................................................................................ 210 Section 9.6 – Complex Solutions to Quadratic Equations .................................................. 214 Lesson 9 Practice Problems .................................................................................................... 217 Lesson 9 Assessment .............................................................................................................. 231 Lesson 5 – Introduction to Exponential Functions ..................................................................... 233 Mini-Lesson 5 ......................................................................................................................... 235 Section 5.1 – Linear Functions vs. Exponential Functions ................................................ 235 Section 5.2 – Characteristics of Exponential Functions ..................................................... 242 Section 5.4 – Applications of Exponential Functions ........................................................ 245 Lesson 5 Practice Problems .................................................................................................... 249 Lesson 5 Assessment .............................................................................................................. 261 Lesson 6 – More Exponential Functions .................................................................................... 263 Mini-Lesson 6 ......................................................................................................................... 265 Section 6.1 – Writing Exponential Models ......................................................................... 265 Section 6.2 – Doubling Time and Halving Time ................................................................ 268 Lesson 6 Practice Problems .................................................................................................... 274 Lesson 6 Assessment .............................................................................................................. 285 Lesson 7 – Logarithms and Logarithmic Functions ................................................................... 287 Mini-Lesson 7 ......................................................................................................................... 289 Section 7.1 – Introduction to Logarithms ........................................................................... 289 Section 7.2 – Computing Logarithms ................................................................................. 292

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Section 7.4 – Solving Logarithmic Equations .................................................................... 297 Section 7.5 – Solving Exponential Equations Algebraically and Graphically ................... 301 Lesson 7 Assessment .............................................................................................................. 318 Lesson 10 – Radical Functions ................................................................................................... 320 Mini-Lesson 10 ....................................................................................................................... 322 Section 10.1 – Roots, Radicals, and Rational Exponents ................................................... 322 Section 10.6 – Solve Radical Equations Algebraically ...................................................... 325 Lesson 10 Practice Problems .................................................................................................. 330 Lesson 10 Assessment ............................................................................................................ 344 Lesson 11 –Rational Functions ................................................................................................... 346 Mini-Lesson 11 ....................................................................................................................... 348 Section 11.2 – Solving Rational Equations......................................................................... 348 Section 11.3 – Applications of Rational Functions ............................................................ 353 Lesson 11 Practice Problems .................................................................................................. 356 Lesson 11 Assessment ............................................................................................................ 368 Lesson 12 – Course Review ........................................................................................................ 370 Mini-Lesson 12 ....................................................................................................................... 372 Section 12.1 – Overview of Functions ................................................................................ 372 Section 12.2 – Solving Equations ....................................................................................... 378 Section 12.3 – Mixed Applications ..................................................................................... 380 Lesson 12 Practice Problems .................................................................................................. 388 Lesson 12 Assessment ............................................................................................................ 418 Appendix A: You Try Answers ................................................................................................. 420

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Lesson 1 – Introduction to Functions Throughout this class, we will be looking at various Algebraic Functions and the characteristics of each. Before we begin, we need to review the concept of what a Function is and look at the rules that a Function must follow. We also need to investigate the different ways that we can represent a Function. It is important that we go beyond simple manipulation and evaluation of these Functions by examining their characteristics analyzing their behavior. Looking at the Functions modeled as Graphs, Tables and Sets of Ordered Pairs is critical to accomplishing that goal.

Lesson Topics: Section 1.1 What is a function?

Definition of function Independent and Dependent Variables

Section 1.2 Multiple Representations of Functions

Sets of ordered pairs (input, output) Tables Graphs Vertical Line Test Behavior of Graphs

Section 1.3 Function Notation

Function evaluation Working with input and output Multiple Representations Graphs and tables with a graphing calculator

Section 1.4 Domain and Range

Definitions Multiple Representations Restricting Domain and Range (calculator)

Section 1.5 Applications of Functions

Criteria for a good graph Practical Domain and Range

Lesson 1 Checklist

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Component

Required? Y or N

Comments

Mini-Lesson

Online Homework

Online Quiz

Online Test

Practice Problems

Lesson Assessment

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Due

Score

Name: _____________________________

Date: ________________

Mini-Lesson 1 Section 1.1 – What is a Function? Intermediate Algebra is a study of functions and their characteristics. In this class, we will study LINEAR, EXPONENTIAL, LOGARITHMIC, QUADRATIC, RATIONAL, & RADICAL functions. Before we learn the specifics of these functions, we need to review/learn the language and notation of FUNCTIONS. What is a Function? The concept of “function” is one that is very important in mathematics. The use of this term is very specific and describes a particular relationship between two quantities: an input quantity and an output quantity. Specifically, a relationship between two quantities can be defined as function if it is the case that “each input value is associated with only one output value”. Why Do We Care About Functions? Imagine that you are a nurse working the emergency room of a hospital. A very sick person arrives. You know just the medicine needed but you are unsure the exact dose. First, you determine the patient’s weight (200 pounds). Then you look at the table to the right and see the given dosage information: Weight in lbs.

mL of Medicine

200

10

200

100

You are immediately confused and very concerned. How much medicine do you give? 10 ml or 100 ml? One amount could be too much and the other not enough. How do you choose the correct amount? What you have here is a situation that does NOT define a function (and would not occur in real life). In this case, for the input value 200 lbs, there are two choices for the output value. If you have a function, you will not have to choose between output values for a given input. In the real case of patients and medicine, the dosage charts are based upon functions.

A More Formal Definition of Function: A FUNCTION is a rule that assigns a single, unique output value to each input value.

Lesson 1 – Introduction to Functions

Problem 1

Mini-Lesson

MEDIA EXAMPLE – Do The Data Represent A Function?

The table below gives the height H, in feet, of a golf ball t seconds after being hit. t = Time (in seconds) 0 1 2 3 4 5 6

H = Height (in feet) 0 80 128 144 128 80 0

a) Identify the input quantity (include units)._________________________________________ Identify the input variable._________________________________________________ Identify the output quantity (include units).________________________________________ Identify the output variable._________________________________________________

b) Write the data as a set of ordered pairs.

c) Interpret the meaning of the ordered pair (3, 144).

d) Is height of the golf ball a function of time? Why or why not?

e) Is time a function of the height of the golf ball? Why or why not?

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Lesson 1 – Introduction to Functions

Problem 2

Mini-Lesson

WORKED EXAMPLE – Investigating Functional Relationships

Let’s investigate the functional relationship between the two quantities, “numerical grade” and “letter grade”. First, let Numerical Grade be the input quantity and Letter Grade be the output quantity. Below is a sample data set that is representative of the situation. Numerical grade 95 92 85 73

Letter Grade A A B C

The numbers above are made up to work with this situation. Other numbers could be used. We are assuming a standard 90, 80, 70, etc… grading scale. Hopefully you can see from this data that no matter what numerical value we have for input, there is only one resulting letter grade. Notice that the repeated outputs “A” are not a problem since the inputs are different. You can uniquely predict the output for any numerical grade input. So, from this information we can say that Letter Grade (output) is a function of Numerical Grade (input). Now let’s switch the data set above. Letter Grade A A B C

Numerical Grade 95 92 85 73

Can you see there is a problem here? If you say that you have an A in a class, can you predict your numerical grade uniquely? No. There are a whole host of numerical scores that could come from having an A. The same is true for all the other letter grades as well. Therefore, Numerical Grade (output) is NOT a function of Letter Grade (input). Summary:

Letter Grade IS a function of Numerical Grade but Numerical Grade is NOT a function of Letter Grade

Additional Terminology In the language of functions, the phrase INDEPENDENT VARIABLE means input and the phrase DEPENDENT VARIABLE means output. The dependent variable (output) “depends on” or is a “function of” the independent variable (input).

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Lesson 1 – Introduction to Functions

Problem 3

Mini-Lesson

YOU TRY – Do The Data Represent A Function?

The table below gives the value of a car n years after purchase n = Time (in years) 0 1 2 3 4 5 6

V = Value (in dollars) 32540 28310 24630 21428 18642 16219 14110

a) Identify the input quantity (include units)._________________________________________

Identify the output quantity (include units).________________________________________

b) Identify the dependent variable._________________________________________________

Identify the independent variable.________________________________________________

c) Interpret the meaning of the ordered pair (2, 24630).

d) Is the value of the car a function of time? Why or why not?

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Lesson 1 – Introduction to Functions

Mini-Lesson

Section 1.2 – Multiple Representations of Functions Problem 4

MEDIA EXAMPLE – Determine Functional Relationships Using Multiple Representations

SETS OF ORDERED PAIRS (input, output) Which of the following represent functional relationships? {(-3, 2), (5, 0), (4, -7)}

{(0, 2), (5, 1), (5, 4)}

{(-3, 2), (5, 2), (4, 2)}

TABLES Which of the following represent functional relationships? x 2 4 4 7

y 52 41 30 19

x 3 11 24 38

y 128 64 16

x 0 1 2 3

y 4 4 4 4

GRAPHS Which of the following represent functional relationships?

THE VERTICAL LINE TEST • If all vertical lines intersect the graph of a relation at only one point, the relation is also a function. One and only one output value exists for each input value. • If any vertical line intersects the graph of a relation at more than one point, the relation “fails” the test and is NOT a function. More than one output value exists for some (or all) input value(s).

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Lesson 1 – Introduction to Functions

Mini-Lesson

WORKED EXAMPLE – Determine Functional Relationships Using Multiple Representations

Problem 5

The table below shows 3 different representations for two relationships. Determine which relationship defines a function.

Set of Ordered Pairs

Table

Functions

Not Functions

{(-7, 6), (-3, 3), (1, 8), (5, 8), (11, 0)}

{(8, 3), (6, 1), (8, -1), (6, 11), (2, -5)}

No input value is repeated in an ordered pair.

Two of the listed input values (6 & 8) are associated with more than one output value.

x y

-4 8

-2 3

0 5

1 3

5 10

x y

0 0

OR x -4 -2 0 1 5

1 1

1 -1

4 2

4 -2

OR y 8 3 5 3 10

x 0 1 1 4 4

y 0 1 -1 2 -2

All input values are associated with one, unique output value.

Two of the listed input values (1 & 4) are associated with more than one output value.

No vertical line intersects the graph in more than one point. We say the graph PASSES the VERTICAL LINE TEST.

Vertical lines intersect the graph at more than one point meaning inputs are repeated with different outputs. We say that the graph FAILS the VERTICAL LINE TEST.

Graph

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Lesson 1 – Introduction to Functions

Mini-Lesson

YOU TRY – Determine Functional Relationships Using Multiple Representations

Problem 6

Which of the following represent functional relationships? A

B x 5 5 5 5

{(4, 1), (7, 1), (-3, 1), (5, 1)}

D

E {(3, 5), (3, 6), (8, 1), (5, 4)}

Problem 7

C y 4 6 8 1 F x 0 3 5 11

y 2 2 3 5

MEDIA EXAMPLE – Does the Statement Describe A Function?

Explain your choice for each of the following. Remember when the word “function” is used, it is in a purely MATHEMATICAL sense, not in an everyday sense. a) Is the number of children a person has a function of their income?

b) Is your weekly pay a function of the number of hours you work each week? (Assume you work at an hourly rate job with no tips).

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Lesson 1 – Introduction to Functions

Problem 8

Mini-Lesson

WORKED EXAMPLE – Behavior of Functions

A function is: INCREASING if the outputs get larger, DECREASING if the outputs get smaller, CONSTANT if the outputs do not change. NOTE: We read graphs just like we read a book…from left to right.

a) The following functions are INCREASING x 0 1 2 3

y 4 6 12 24

x 0 1 2 3

y 10 5 0 -5

x 0 1 2 3

y 4 4 4 4

b) The following functions are DECREASING

c) The following functions are CONSTANT

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Lesson 1 – Introduction to Functions

Mini-Lesson

Section 1.3 – Function Notation FUNCTION NOTATION is used to indicate a functional relationship between two quantities as follows: Function Name (INPUT) = OUTPUT So, the statement f (x) = y would refer to the function f , and correspond to the ordered pair (x,y), where x is the input variable, and y is the output variable.

Function Evaluation: To evaluate a function at a particular value of the input variable, replace each occurrence of the input variable with the given value and compute the result. Note: Use of ( ) around your input value, especially if the input is negative, can help achieve correct results.

Problem 9

MEDIA EXAMPLE – Function Evaluation

Given f (x) = 2x – 5, evaluate f (2), f (–1), f (x + 1) and f (–x).

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Lesson 1 – Introduction to Functions

Problem 10

Mini-Lesson

WORKED EXAMPLE – Function Evaluation

If f (x) = 5x2 – 3x – 10, find f (2) and f (–1). f (2) = 5(2)2 – 3(2) – 10 = 5(4) – 6 – 10 = 20 – 6 – 10 = 14 – 10 =4

f (–1) = 5(–1)2 – 3(–1) – 10 = 5(1) + 3 – 10 = 5 + 3 – 10 = 8 – 10 = –2

When working with FUNCTIONS, there are two main questions we will ask and solve as follows: Given a particular INPUT value, what is the corresponding OUTPUT value? Given a particular OUTPUT value, what is the corresponding INPUT value? Problem 11

MEDIA EXAMPLE – Working with Input and Output

Given f (x) = 2x + 5, determine each of the following. Write your answers as ordered pairs. GIVEN INPUT FIND OUTPUT Find f (0)

Find f (–2)

GIVEN OUTPUT FIND INPUT Find x if f (x) = 7

Find x if f (x) = –11

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Lesson 1 – Introduction to Functions

Problem 12

Mini-Lesson

YOU TRY – Working with Input and Output

Given f (x) = –3x – 4, compute each of the following. Show all steps, and write your answers as ordered pairs. Write answers as integers or reduced fractions (no decimals). a) Find f (2)

b) Find x if f (x) = 7

c) Find f (–3)

d) Find x if f (x) = –12

e) Find f (–x)

f) Find f (x – 5)

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Lesson 1 – Introduction to Functions

Problem 13

Mini-Lesson

MEDIA EXAMPLE – Working with Function Notation Using a Set of Ordered Pairs

The function g(x) is shown below g = {(1, 3), (5, 2), (8, 3), (6, -5)} g(1) = __________

Find x if g(x) = -5. x = __________

Find x if g(x) = 3. x = _____________________

Problem 14

MEDIA EXAMPLE – Working with Function Notation Using a Table

The function V(n) is shown below gives the value, V, of an investment (in thousands of dollars) after n months. n 1 2 3 4 V(n) 2.31 3.02 5.23 3.86

Identify the input quantity (include units).________________________________ Identify the output quantity (include units).________________________________ Write a sentence explaining the meaning of the statement V(1) = 2.31.

Determine V(3) and write a sentence explaining its meaning.

For what value of n is V(n) = 3.02? Interpret your answer in a complete sentence.

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Lesson 1 – Introduction to Functions

Problem 15

Mini-Lesson

MEDIA EXAMPLE – Working with Function Notation Using a Graph

The function D(t) below shows a person’s distance from home as a function of time.

Identify the input quantity (include units).________________________________ Identify the output quantity (include units).________________________________ Write a sentence explaining the meaning of the statement D(15) = 10.

Determine D(0) and write a sentence explaining its meaning.

For what value of t is D(t) = 0? Interpret your answer in a complete sentence.

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Lesson 1 – Introduction to Functions

Problem 16

Mini-Lesson

EXAMPLE – Create a table and Graph of a Function

Consider the function y = 5 – 2x a) Complete the table below x

0

3

7

y b) Sketch the graph of y = 5 – 2x.

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9

12

Lesson 1 – Introduction to Functions

Mini-Lesson

Section 1.4 – Domain and Range The DOMAIN of a function is the set of all possible values for the input quantity. The RANGE of a function is the set of all possible values for the output quantity

Problem 17

MEDIA EXAMPLE – Domain and Range, Multiple Representations

SET OF ORDERED PAIRS Determine the domain and range of the function P(x) = {(2, 3), (4, –5), (6, 0), (8, 5)} Domain: ________________________________________________ Range: _________________________________________________

TABLE Determine the domain and range of the function R(t) defined below. t 0 2 5 8 R(t) 23 54 66 87

11 108

Domain: ________________________________________________ Range: _________________________________________________

GRAPH Determine the domain and range of the function g(x) defined below. Domain of g(x) Inequality Notation

Range of g(x): Inequality Notation

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Interval Notation

Interval Notation

Lesson 1 – Introduction to Functions

Mini-Lesson

YOU TRY – Domain and Range, Multiple Representations

Problem 19

Find the domain and range for the functions below. Use proper notation for your domain/range responses. a) Set of ordered pairs D(r) = {(7, 8), (8, 12), (11, 21)}

Domain: ____________________________

Range: _____________________________

b) Table of values n 3 6 8

A(n) 51 42 33

Domain: ____________________________

Range: _____________________________

c) Graph Domain of f(x) Inequality Notation

Interval Notation

Range of f(x): Inequality Notation

Interval Notation

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Lesson 1 – Introduction to Functions

Mini-Lesson

Section 1.5 – Applications of Functions Criteria for a GOOD GRAPH: 1. The horizontal axis should be properly labeled with the name and units of the input quantity. 2. The vertical axis should be properly labeled with the name and units of the output quantity. 3. Use an appropriate scale. Start at or just below the lowest value. End at or just above the highest value. Scale the graph so the adjacent tick marks are equal distance apart. Use numbers that make sense for the given data set. The axes meet at (0,0) Use a “//” between the origin and the first tick mark if the scale does not begin at 0. 4. All points should be plotted correctly, and the graph should be neat and uncluttered.

Problem 20

MEDIA EXAMPLE – Understanding Applications of Functions

Suppose that the cost to fill your 15-gallon gas tank is determined by the function C(g) = 3.29g where C is the output (cost in $) and g is the input (gallons of gas). a) Draw a GOOD graph of this function in the space below. Provide labels for your axes.

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Lesson 1 – Introduction to Functions

Mini-Lesson

b) Identify the first and last ordered pairs that are on the graph (based on the information above). Include both ordered pairs and function notation.

c) What is the INPUT quantity (including units) for this function? Name the smallest and largest possible input quantity then use this information to identify the PRACTICAL DOMAIN.

d) What is the OUTPUT quantity (including units) for this function? Name the smallest and largest possible output quantity then use this information to identify the PRACTICAL RANGE.

Practical Domain: The PRACTICAL DOMAIN of a function is the set of all possible input values that are realistic for a given problem. Practical Range: The PRACTICAL RANGE of a function is the set of all possible output values that are realistic for a given problem.

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Lesson 1 – Introduction to Functions

Problem 21

Mini-Lesson

WORKED EXAMPLE – Practical Domain and Range

Let the function M(t) = 15t represent the distance you would travel bicycling t hours. Assume you can bike no more than 10 hours. Find the practical domain and practical range for this function.

BEGIN by drawing an accurate graph of the situation. Try and determine the smallest and largest input values then do the same thing for the output values.

PRACTICAL DOMAIN In this situation, the input values you can use are related to biking and the input is TIME. You are told you can bike no more than 10 hours. You also cannot bike a negative number of hours but you CAN bike 0 hours.

PRACTICAL RANGE In this situation, the outputs represent distances traveled depending on how long you bike. Looking at the endpoints for Practical Domain, you can find you Practical Range as follows:

Therefore, the Practical Domain is

M (0) M (t ) M (10)

0 t 10 hours

Thus, 0 M (t ) 150 miles is your Practical Range

This means “all the values of t between and including 0 and 10”.

This means you can bike a minimum of 0 miles and a maximum of 150 miles in this situation.

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Lesson 1 – Introduction to Functions

Problem 22

Mini-Lesson

YOU TRY – Applications of Functions

A local towing company charges $3.25 per mile driven plus a base fee of $30.00. They tow a maximum of 25 miles. a) Let C represent the total cost of any tow and x represent miles driven. Using correct and formal function notation, write a function that represents total cost as a function of miles driven.

b) Identify the practical domain of this function by filling in the blanks below. Minimum miles towed ≤ x ≤ Maximum miles towed Practical Domain: _____________ ≤ x ≤ ______________ c) Identify the practical range of this function by filling in the blanks below. Minimum Cost ≤ C(x) ≤ Maximum Cost Practical Range: _____________ ≤ C(x) ≤ ______________ d) Write a complete sentence to explain the meaning of C(60) = 225 in words.

e) Use your function from part a) to find C(15). Write your answer as ordered pair then explain its meaning in a complete sentence.

f) Use your function from part a) to determine the value of x when C(x) = 30. Write your answer as ordered pair then explain its meaning in a complete sentence.

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Lesson 1 – Introduction to Functions

Problem 23

Mini-Lesson

YOU TRY – Applications of Functions

The value V (in dollars) of a washer/dryer set decreases as a function of time t (in years). The function V(t) = –100t + 1200 models this situation. You own the washer/dryer set for 12 years. a) Identify the input quantity (including units) and the input variable.

b) Identify the output quantity (including units) and the output variable.

c) Fill in the table below. t

0

6

12

V(t)

d) Draw a GOOD graph of this function in the space below. Provide labels for your axes. Plot and label the ordered pairs from part c).

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Lesson 1 – Introduction to Functions

Mini-Lesson

e) A washer/dryer set that is worth $400 would be how old? Hint: This is a GIVEN OUTPUT FIND INPUT question. You must show work.

f) After 2 years, how much would the washer/dryer set be worth? Hint: This is a GIVEN INPUT FIND OUTPUT question. You must show work.

g) What is the practical domain for V(t)?

Inequality notation: ____________________________________________

Interval notation: ____________________________________________

h) What is the practical range for V(t)?

Inequality notation: ____________________________________________

Interval notation: ____________________________________________

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Name: _____________________________

Date: ________________

Lesson 1 Practice Problems Section 1.1: What is a Function? 1. The table below gives the distance D, in kilometers, of a GPS satellite from Earth t minutes after being launched. t = Time (in minutes) D = Distance (in km) 0 0 20 4003 40 9452 60 14,232 80 18,700 100 20,200 120 20,200 a) Identify the input quantity (include units)._____________________________________ Identify the input variable._________________________________________________ Identify the output quantity (include units).____________________________________ Identify the output variable._________________________________________________ b) Write the data as a set of ordered pairs.

c) Interpret the meaning of the ordered pair (40, 9452).

d) Is distance of the satellite a function of time? Why or why not?

e) Is time a function of the distance of the satellite from Earth? Why or why not?

Lesson 1 – Introduction to Functions

Assessment

2. The table below gives the number of Gene copies, G, t minutes after observation. t = Time (in minutes) 0 3 5 6 8 10 12

G = number of Gene Copies 52 104 165 208 330 524 832

a) Identify the input quantity (include units)._____________________________________ Identify the input variable._________________________________________________ Identify the output quantity (include units).____________________________________ Identify the output variable._________________________________________________

b) Write the data as a set of ordered pairs.

c) Interpret the meaning of the ordered pair (6, 208).

d) Is the number of Gene copies a function of time? Why or why not?

e) Is time a function of the number of Gene copies? Why or why not?

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Lesson 1 – Introduction to Functions

Assessment

3. The table below gives the number of homework problems, H, that Tara has completed t minutes after she began her homework. t = Time (in minutes) 0 10 20 30 40 50 60

H = number of homework problems completed 0 3 8 8 15 17 20

a) Identify the input quantity (include units)._____________________________________ Identify the input variable._________________________________________________ Identify the output quantity (include units).____________________________________ Identify the output variable._________________________________________________

b) Write the data as a set of ordered pairs.

c) Interpret the meaning of the ordered pair (40, 15).

d) Is the number of homework problems completed a function of time? Why or why not?

e) Is time a function of the number of homework problems completed? Why or why not?

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Lesson 1 – Introduction to Functions

Assessment

4. The table below gives the number of hot dogs, H, that a competitive hot dog eater has eaten t minutes after the start of the competition. t = Time (in minutes) 0 1 3 5 7 9 10

H = number of hotdogs eaten 0 8 23 37 50 63 68

a) Identify the input quantity (include units)._____________________________________ Identify the input variable._________________________________________________ Identify the output quantity (include units).____________________________________ Identify the output variable._________________________________________________

b) Write the data as a set of ordered pairs.

c) Interpret the meaning of the ordered pair (7, 50).

d) Is the number of hot dogs eaten a function of time? Why or why not?

e) Is time a function of the number of hot dogs eaten? Why or why not?

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Lesson 1 – Introduction to Functions

Assessment

Section 1.2: Multiple Representations of Functions 5. Determine whether the following sets of ordered pairs represent a functional relationship. Justify your answer. a) R = {(2, 4), (3, 8), (–2, 6)} b) T = {(3, –2), (4, –1), (5, 8), (3, –2)} c) L = {(3, –5), (1, –2), (2, –2), (3, 5)} d) A = {(5, –5), (6, –5), (7, –5} e) F = {(2, –3), (6,

), (4, 8)}

6. Determine whether the following tables of values represent a functional relationship. Justify your answer.

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Lesson 1 – Introduction to Functions

Assessment

7. Determine whether the following graphs represent a functional relationship. Justify your answer.

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Lesson 1 – Introduction to Functions

Assessment

8) Determine whether the following scenarios represent functions. Explain your choice for each of the following. Remember when the word “function” is used, it is in a purely MATHEMATICAL sense, not in an everyday sense. a) Is a person’s height a function of their age? b) Is a person’s age a function of their date of birth?

c) Is the growth of a tree a function of the monthly rainfall?

d) John says that time he will spend on vacation will be determined by the number of overtime hours he works on his job. Is it true that his vacation time is a function of his overtime hours?

e) Sara says that the number of tomatoes she grows will be determined by the weather. Is it true that the size of his tomato crop is a function of the weather?

9. Determine whether the following functions represented by the graphs below are increasing, decreasing, or constant.

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Lesson 1 – Introduction to Functions

Assessment

10. Determine whether the following functions represented by the tables below are increasing, decreasing, or constant. a)

x -2 -1 0 1 2

f(x) 5 8 23 37 49

b)

t 4 7 12 13 17

s(t) 5 3 -8 -12 -25

d)

x h(x)

0 2

1 2

2 2

3 2

e)

x h(x)

-5 27

1 26

8 24

9 23

f)

x h(x)

0 -22

1 -20

2 -18

3 -15

Section 1.3: Function Evaluation 11. Given the function 𝑓(𝑥) = −𝑥 + 6, evaluate each of the following a) 𝑓(2) =

b) 𝑓(−1) =

c) 𝑓(0) =

Page 39

c)

x 0 1 2 3 4

g(x) 5 5 5 5 5

Lesson 1 – Introduction to Functions

Assessment

12. Given the function 𝑠(𝑡) = 14 − 2𝑡, evaluate each of the following: a) 𝑠(−3) =

b) 𝑠(4) =

c) 𝑠(0) =

13. Given the function ℎ(𝑐) = 2𝑐 2 − 3𝑐 + 4, evaluate each of the following: a) ℎ(−2) =

b) ℎ(3) =

c) ℎ(0) =

14. Given the function 𝑔(𝑥) = −𝑥 2 + 3𝑥, evaluate each of the following: a) 𝑔(−3) =

b) 𝑔(4) =

c) 𝑔(0) =

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Lesson 1 – Introduction to Functions

Assessment

15. Given the function 𝑓(𝑥) = −𝑥 + 6, evaluate each of the following a) 𝑓(2𝑥) =

1

b) 𝑓 (2 𝑥) =

c) 𝑓(𝑥 − 3) =

16. Given the function 𝑠(𝑡) = 14 − 2𝑡, evaluate each of the following: a) 𝑠(3𝑡) =

1

b) 𝑠 (4 𝑡) =

c) 𝑠(𝑡 + 4) =

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Lesson 1 – Introduction to Functions

Assessment

17. Given the function ℎ(𝑐) = 2𝑐 2 − 3𝑐 + 4, evaluate each of the following: a) ℎ(−2𝑐) =

b) ℎ(𝑐 − 1) =

c) ℎ(𝑥 + 2) =

18. Given 𝑓(𝑥) = 3𝑥 − 6, determine each of the following. Also determine if you are given an input or output and whether you are finding an input or output and write your result as an ordered pair. a) Find 𝑓(2) = b) Find x if 𝑓(𝑥)= 3

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

c) Find 𝑓(−4) =

d) Find x if 𝑓(𝑥) = −12

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

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Lesson 1 – Introduction to Functions

Assessment

Ordered pair:______________________ 3

Ordered pair:______________________

1

19. Given 𝑔(𝑥) = 2 𝑥 − 2, determine each of the following. Write your final result as a fraction when appropriate. a) Find 𝑔(4) =

b) Find x if 𝑔(𝑥)= 3

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________ 7

c) Find 𝑔(−8) =

d) Find x if 𝑔(𝑥) = − 2

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:_____________________

Ordered pair:______________________

20. Use the table below to find the function values. t k(t)

−8 14

−3 7

2 0

a) 𝑘(7) = 𝑏) 𝑘(−3) =

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7 −7

12 −14

Lesson 1 – Introduction to Functions

Assessment

𝑐) 𝑘(−8) = 21. Given the table for the function below, determine each of the following. Also determine if you are given an input or output and whether you are finding an input or output and write your result as an ordered pair. x f(x)

−6 9

−4 3

−2 −5

a) Find x if 𝑓(𝑥) = −12

0 −12

2 −17

b) Find 𝑓(−4) =

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

c) Find x if 𝑓(𝑥)= 3

d) Find 𝑓(2) =

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

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Lesson 1 – Introduction to Functions

Assessment

22. Given the graph for the function below, determine each of the following. Also determine if you are given an input or output and whether you are finding an input or output and write your result as an ordered pair.

a) Find x if 𝑓(𝑥) = 5

b) Find 𝑓(−2) =

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

c) Find x if 𝑓(𝑥)= 3

d) Find 𝑓(3) =

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

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Lesson 1 – Introduction to Functions

Assessment

23. Given the graph for the function below, determine each of the following. Also determine if you are given an input or output and whether you are finding an input or output and write your result as an ordered pair.

a) Find any x-values where 𝑔(𝑥) = 5

b) Find 𝑔(2) =

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

c. Find any x-values where 𝑔(𝑥) = 0

d) Find 𝑔(3) =

Given input or output?______________

Given input or output?______________

Finding input or output?_____________

Finding input or output?_____________

Ordered pair:______________________

Ordered pair:______________________

Page 46

Lesson 1 – Introduction to Functions

Assessment

24. Consider the function 𝑦 = 2𝑥 − 3 a) Complete the table below x

−3

−1

0

1

3

0

1

3

y b) Sketch the graph of 𝑦 = 2𝑥 − 3

25. Consider the function 𝑓(𝑥) = −3𝑥 + 4 a) Complete the table below x

−3

−1

y b) Sketch the graph of 𝑓(𝑥) = −3𝑥 + 4

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Lesson 1 – Introduction to Functions

Assessment

Section 1.4: Domain and Range 26. For each set of ordered pairs, determine the domain and the range. a) g = {(3, –2), (5, –1), (7, 8), (9, –2), (11, 4), (13, –2) } Domain: Range: b) f = {(–2, –5), (–1, –5), (0, –5), (1, –5)} Domain: Range: c) h = {(–3, 2), (1, –5), (0, –3), (4, –2) } Domain: Range:

27. For each table of values, determine the domain and range of the function. a)

b)

x –10 –5 0 5 10

f(x) 3 8 12 15 18

Domain:

x –20 –10 0 10 20 30

g(x) –4 14 32 50 68 86

Domain:

Range:

Range:

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Lesson 1 – Introduction to Functions

Assessment

c) H T(h)

8 54

9 62

10 66

11 69

12 72

1 73

2 74

3 73

4 72

Domain:

Range:

28. For each graph, determine the domain and range of the function. Use inequality and interval notation when appropriate.

Domain:

Domain:

Inequality notation:_________________

Inequality notation:_________________

Interval notation:___________________

Interval notation:___________________

Range:

Range:

Inequality notation:_________________

Inequality notation:_________________

Interval notation:___________________

Interval notation:___________________

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Lesson 1 – Introduction to Functions

Assessment

Domain:

Domain:

Inequality notation:_________________

Inequality notation:_________________

Interval notation:___________________

Interval notation:___________________

Range:

Range:

Inequality notation:_________________

Inequality notation:_________________

Interval notation:___________________

Interval notation:___________________

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Lesson 1 – Introduction to Functions

Assessment

Section 1.5: Applications of Functions 29. A local window washing company charges $0.50 per window plus a base fee of $20.00 per appointment. They can wash a maximum of 200 windows per appointment.

a) Let C represent the total cost of an appointment and w represent the number of windows washed. Using correct and formal function notation, write a function that represents total cost as a function of windows washed.

b) Identify the practical domain of this function by filling in the blanks below. Minimum windows washed ≤ w ≤ Maximum windows washed Practical Domain: _____________ ≤ w ≤ ______________ c) Identify the practical range of this function by filling in the blanks below. Minimum Cost ≤ C(w) ≤ Maximum Cost Practical Range: _____________ ≤ C(w) ≤ ______________ d) Enter the equation for C into the Y= part of your calculator. Then use the TABLE feature to complete the table below: w

0

50

150

200

C(w)

e) Find the value of C(50). Circle the appropriate column in the table. Write a sentence explaining the meaning of your answer.

g) Use your function from part a) to determine the value of w when C(w) = 45. Set up the equation, C(w) = 45 then solve for the value of w.

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Lesson 1 – Introduction to Functions

Assessment

30. Suppose the number of pizzas you can make in an 8 hour day is determined by the function P(t) = 12t where P is the output (Pizzas made) and t is the input (Time in hours). a) Graph this function using a few sample points. Example: using t=2, we get P(t)= 24, so the point (2, 24) is on the graph. Show a good graph in the space below.

b) Use your graph to identify the first and last ordered pairs that are on the graph (based on the information above). Include both ordered pairs and function notation.

c) What is the INPUT quantity (including units) for this function? Name the smallest and largest possible input quantity then use this information to identify the PRACTICAL DOMAIN. Input quantity (including units):_______________________________ Practical domain: Inequality notation:_______________________ Interval notation:_________________________

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Lesson 1 – Introduction to Functions

Assessment

d) What is the OUTPUT quantity (including units) for this function? Name the smallest and largest possible output quantity then use this information to identify the PRACTICAL RANGE. Output quantity (including units):_______________________________ Practical range: Inequality notation:_______________________ Interval notation:_________________________

e) Find P(3) and interpret its meaning in the context of the problem.

f) Find t so that P(t) = 70 and interpret its meaning in the context of the problem.

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Lesson 1 – Introduction to Functions

Assessment

31. The life expectancy for males in the United States from the year 1900 until 2020 can be modeled by the function L(x) = 0.27x + 48.3, where L is the life expectancy and x is the number of years since 1900. a) Which letter, L or x is used for input?

b) What does the INPUT represent? Include units.

c) Which letter, L or x, is used for output?

d) What does the OUTPUT represent? Include units.

e) Draw a neat, labeled and accurate sketch of this graph in the space below. x

L(x)

0 20 40 60 80 100 120

f) What is the practical domain of L(x)? Use proper inequality notation.

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Lesson 1 – Introduction to Functions

Assessment

g) What is the practical range of L(x)? Use proper inequality notation.

h) What is the life expectancy of a man born in Iowa in 1950?

i) If a man is expected to live to the age of 60, approximate the year he was born. (Round to one decimal place)?

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Lesson 1 – Introduction to Functions

Assessment

Lesson 1 Assessment 1. Let r(a) = 4 – 5a. Show all steps. Write each answer using function notation and as an ordered pair. a) Determine r(–2). b) For what value of a is r(a) = 19?

2. The graph of f(x) is given below. Use inequality notation. a) Give the domain of f(x):

b) Give the range of f(x):

c) f (0) = _________

d) f (x) = 0 when x =_________________

3. Consider the following table of values. Fill in the blanks below, and identify the corresponding ordered pairs. x g(x)

–2 1

g(1) = _______,

–1 4

0 2

1 6

g(x) = 1 when x = ________,

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2 5

3 0

4 2

g(x) = 2 when x = _________________

Lesson 1 – Introduction to Functions

Assessment

4. The height, h (in feet), of a golf ball is a function of the time, t (in seconds), it has been in flight. A golfer strikes the golf ball with an initial upward velocity of 96 feet per second. The maximum height of the ball is 144 feet. The height of the ball above the ground is given by the function h(t) = –16t2 + 96t. a) Complete the table below. t 0 1

2

3

4

5

h(t) b) Determine h(3) . Write a sentence explaining the meaning of your answer.

c) For what values of t is h(t) = 0? Explain the meaning of your answers.

d) Determine the practical domain. Use inequality notation and include units.

e) Determine the practical range. Use inequality notation and include units.

f) Use your table to generate a graph of h(t). Use the practical domain and range to determine a “good” viewing window.

Page 57

6

Lesson 2 – Functions and Function Operations As we continue to work with more complex functions it is important that we are comfortable with Function Notation, operations on Functions and operations involving more than one function. In this lesson, we study using proper Function Notation and then spend time learning how add, subtract, multiply and divide Functions, both algebraically and when the functions are represented with a tables or graphs. Finally, we take a look at a couple of real world examples that involve operations on functions. Lesson Topics: Section 2.1: Combining Functions

Basic operations: Addition, Subtraction, Multiplication, and Division Multiplication Property of Exponents Division Property of Exponents Negative Exponents Operations on Functions in table form Operations on Functions in graph form

Section 2.2: Applications of Function Operations

Cost, Revenue, and Profit

Section 2.3: Composition of Functions

Evaluating Functions Composition of Functions in table form Composition of Functions in graph form

Section 2.4: Applications of Function Composition

Page 58

Lesson 2 Checklist Component

Required? Y or N

Comments

Mini-Lesson

Online Homework

Online Quiz

Online Test

Practice Problems

Lesson Assessment

Page 59

Due

Score

Name: _____________________________

Date: ________________

Mini-Lesson 2 Section 2.1 – Combining Functions Function notation can be expanded to include notation for the different ways we can combine functions as described below. Basic Mathematical Operations The basic mathematical operations are: addition, subtraction, multiplication, and division. When working with function notation, these operations will look like this: Addition

Subtraction

Multiplication

f (x) + g(x)

f ( x ) g ( x)

f ( x) g ( x)

Division f (x) g(x) ≠ 0 g(x)

Many of the problems we will work in this lesson are problems you may already know how to do. You will just need to get used to some new notation. We will start with the operations of addition and subtraction. Problem 1 WORKED EXAMPLE – Adding and Subtracting Functions Given f (x) = 2x2 + 3x – 5 and g(x) = –x2 + 5x + 1. a) Find f (x) + g(x)

b) Find f (x) – g(x)

c) Find f (1) – g(1)

𝑓 (𝑥) + 𝑔(𝑥) = (2𝑥 2 + 3𝑥 − 5) + (−𝑥 2 + 5𝑥 + 1) = 2𝑥 2 + 3𝑥 − 5 − 𝑥 2 + 5𝑥 + 1 = 2𝑥 2 − 𝑥 2 + 3𝑥 + 5𝑥 − 5 + 1 𝑓(𝑥) + 𝑔(𝑥) = 𝑥 2 + 8𝑥 − 4 𝑓 (𝑥) − 𝑔(𝑥) = (2𝑥 2 + 3𝑥 − 5) − (−𝑥 2 + 5𝑥 + 1) = 2𝑥 2 + 3𝑥 − 5 + 𝑥 2 − 5𝑥 − 1 = 2𝑥 2 + 𝑥 2 + 3𝑥 − 5𝑥 − 5 − 1 𝑓(𝑥) − 𝑔(𝑥) = 3𝑥 2 − 2𝑥 − 6 𝑓 (1) − 𝑔(1) = [2(1)2 + 3(1) − 5] − [−(1)2 + 5(1) + 1] = (2 + 3 − 5) − (−1 + 5 + 1) = 0−5 𝑓(1) − 𝑔(1) = −5

Lesson 2 – Functions and Function Operations

Assessment

Problem 2 MEDIA EXAMPLE – Adding and Subtracting Functions Given f (x) = 3x2 + 2x – 1 and g(x)= x2 + 2x + 5: a) Find f (x) + g(x)

b) Find f (x) – g(x)

Problem 3 YOU TRY – Adding and Subtracting Functions Given f (x) = x2 + 4 and g(x)= x2 + 1, determine each of the following. Show complete work. a) Find f (2) + g(2)

b) Find f (x) – g(x)

c) Find f (2) – g(2)

Page 61

Lesson 2 – Functions and Function Operations

Assessment

Function Multiplication and the Multiplication Property of Exponents When multiplying functions, you will often need to work with exponents. The following should be familiar to you and will come into play in the examples below: MULTIPLICATION PROPERTY OF EXPONENTS Let m and n be rational numbers. To multiply powers of the same base, keep the base and add the exponents: 𝒂𝒎 · 𝒂𝒏 = 𝒂𝒎+𝒏 Problem 4 WORKED EXAMPLE – Function Multiplication a) Given 𝑓(𝑥) = −8𝑥 4 and 𝑔(𝑥) = 5𝑥 3 , find 𝑓(𝑥) · 𝑔(𝑥) 𝑓(𝑥) · 𝑔(𝑥) = (−8𝑥 4 )(5𝑥 3 ) = (−8)(5)(𝑥 4 )(𝑥 3 ) = (−40)(𝑥 4+3 ) 𝑓(𝑥) · 𝑔(𝑥) = −40𝑥 7

Reorder using Commutative Property Simplify using the Multiplication Property of Exponents Final Result

b) Given 𝑓(𝑥) = −3𝑥 and 𝑔(𝑥) = 4𝑥 2 − 𝑥 + 8, find 𝑓(𝑥) · 𝑔(𝑥) 𝑓(𝑥) · 𝑔(𝑥) = (−3𝑥)(4𝑥 2 − 𝑥 + 8) = (−3𝑥)(4𝑥 2 ) + (−3𝑥)(−𝑥) + (−3𝑥)(8) 𝑓(𝑥) · 𝑔(𝑥) = −12𝑥 3 + 3𝑥 2 − 24𝑥

Apply the Distributive Property Remember the rules of exp. (−3𝑥)(4𝑥 2 ) = (−3)(4)(𝑥1 )(𝑥 2 ) = −12𝑥 3 Final Result

c) Given f (x) = 3x + 2 and g(x) = 2x – 5, find 𝑓(𝑥) · 𝑔(𝑥) 𝑓(𝑥) · 𝑔(𝑥) = (3𝑥 + 2)(2𝑥 − 5) = (3𝑥)(2𝑥) + (3𝑥)(−5) + (2)(2𝑥) + (2)(−5) = 𝐹𝐼𝑅𝑆𝑇 + 𝑂𝑈𝑇𝐸𝑅 + 𝐼𝑁𝑁𝐸𝑅 + 𝐿𝐴𝑆𝑇 = (6𝑥 2 ) + (−15𝑥) + (4𝑥) + (−10) 𝑓(𝑥) · 𝑔(𝑥) = 6𝑥 2 − 11𝑥 − 10

Page 62

Use FOIL Remember the rules of exp. (3𝑥)(2𝑥) = (3)(2)(𝑥)(𝑥) = 6𝑥 2 Combine Like Terms Final Result

Lesson 2 – Functions and Function Operations

Problem 5 MEDIA EXAMPLE – Function Multiplication Given 𝑓(𝑥) = 3𝑥 + 2 and 𝑔(𝑥) = 2𝑥 2 + 3𝑥 + 1, find 𝑓(𝑥) · 𝑔(𝑥)

Problem 6 YOU TRY – Function Multiplication For each of the following, find 𝑓(𝑥) · 𝑔(𝑥) a) f (x) = 3x – 2 and g(x) = 3x + 2

b) f (x) = 2x2 and g(x) = x3 – 4x + 5

c) f (x) = 4x3 and g(x) = –6x

Page 63

Assessment

Lesson 2 – Functions and Function Operations

Assessment

Function Division and the Division Property of Exponents When dividing functions, you will also need to work with exponents of different powers. The following should be familiar to you and will come into play in the examples below: DIVISION PROPERTY OF EXPONENTS Let m, n be rational numbers. To divide powers of the same base, keep the base and subtract the exponents.

am a m n where a 0 n a

Problem 7 WORKED EXAMPLE – Function Division For each of the following, find f (x) . Use only positive exponents in your final answer. g(x) a) 𝑓(𝑥) = 15𝑥15 and 𝑔(𝑥) = 3𝑥 9

f ( x) 15 x15 g ( x) 3x 9 5 x159 5x6

NEGATIVE EXPONENTS

If a 0 and n is a rational number, then a n

b) 𝑓(𝑥) = −4𝑥 5 and 𝑔(𝑥) = 2𝑥 8

f ( x) 4 x 5 g ( x) 2 x8 2 x 58 2 x 3

Page 64

2 x3

1 an

Lesson 2 – Functions and Function Operations

Problem 8

Assessment

MEDIA EXAMPLE – Function Division

For each of the following, determine f (x) . Use only positive exponents in your final answer. g(x) a) f (x) = 10x4 + 3x2 and g(x) = 2x2

b) f (x) = –12x5 + 8x2 + 5 and g(x) = 4x2

YOU TRY – Function Division For each of the following, determine f (x) . Use only positive exponents in your final answer. g(x) Problem 9

a) f (x) = 25x5 – 4x7 and g(x) = –5x4

b) f (x) = 20x6 – 16x3 + 8 and g(x) = –4x3

Page 65

Lesson 2 – Functions and Function Operations

Assessment

Functions can be presented in multiple ways including: equations, data sets, graphs, and applications. If you understand function notation, then the process for working with functions is the same no matter how the information if presented. MEDIA EXAMPLE – Working with Functions in Table Form

Problem 10

Functions f (x) and g(x)are defined in the tables below. Find a – e below using the tables. x f (x)

–3 8

–2 6

0 3

1 2

4 5

5 8

8 11

10 15

12 20

x g(x)

0 1

2 3

3 5

4 10

5 4

8 2

9 0

11 –2

15 –5

a)

f (1) =

b)

g(9) =

c)

f (0) + g(0) =

d)

g(5) – f(8) =

e)

𝑓(0) · 𝑔(3) =

Problem 11

YOU TRY – Working with Functions in Table Form

Given the following two tables, complete the third table. Show work in the table cell for each column. The first one is done for you. x f (x)

0 4

1 3

2 –2

3 0

4 1

x g(x)

0 6

1 –3

2 4

3 –2

4 2

x

0

1

2

3

4

f (x)+ g(x)

𝑓(0) + 𝑔(0)

𝑓(1) + 𝑔(1)

=4+6 = 10

Page 66

Lesson 2 – Functions and Function Operations

Assessment

If you remember that graphs are just infinite sets of ordered pairs and if you do a little work ahead of time (as in the example below) then the graphing problems are a lot easier to work with. Problem 12

YOU TRY – Working with Functions in Graph Form

Use the graph to determine each of the following. Assume integer answers. The graph of g is the graph in bold.

Complete the following ordered pairs from the graphs above. Use the information to help you with the problems below. The first ordered pair for each function has been completed for you. f: (–7, 2), (–6, (3,

), (4,

g: (–7, 3), (–6, (3,

), (4,

), (–5, ), (5,

), (–4, ), (6,

), (–5,

), (–3,

), (7,

), (–4,

), (5, ), (6,

), (–2,

), (–1,

), (0,

), (1,

), (2,

),

), (–2,

), (–1,

), (0,

), (1,

), (2,

),

)

), (–3,

), (7,

)

a) g(4) = ________________________

b) f(2) =____________________

c) g(0) = ________________________

d) f(-6) = ____________________

e) If f (x) = 0, x = ____________________

f) If g(x) = 0, x = ____________________

g) If f (x) = 1, x =____________________

h) If g(x) = –4, x =____________________

i) f(–1) + g(–1) =____________________

j) g(–6) – f(–6) =____________________

k) 𝑓(1) ∗ 𝑔(−2) =____________________

l)

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g (6) ____________________ f (1)

Lesson 2 – Functions and Function Operations

Assessment

Section 2.2 – Applications of Function Operations One of the classic applications of function operations is the forming of the Profit function, P(x) by subtracting the cost function, C(x), from the revenue function, R(x) as shown below. Profit = Revenue – Cost Given functions P(x) = Profit,

R(x) = Revenue,

and C(x) = Cost:

𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥) Problem 13

MEDIA EXAMPLE – Cost, Revenue, Profit

A local courier service estimates its monthly operating costs to be $1500 plus $0.85 per delivery. The service generates revenue of $6 for each delivery. Let x = the number of deliveries in a given month. a) Write a function, C(x), to represent the monthly costs for making x deliveries per month.

b) Write a function, R(x), to represent the revenue for making x deliveries per month.

c) Write a function, P(x), that represents the monthly profits for making x deliveries per month.

d) Determine algebraically the break-even point for the function P(x) and how many deliveries you must make each month to begin making money. Show complete work. Write your final answer as a complete sentence.

e) Determine the break-even point graphically by solving the equation P(x) = 0. Explain your work and show the graph with appropriate labels. Write your final answer as a complete sentence.

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Lesson 2 – Functions and Function Operations

Assessment

Problem 14 YOU TRY – Cost, Revenue, Profit February is a busy time at Charlie’s Chocolate Shoppe! During the week before Valentine’s Day, Charlie advertises that his chocolates will be selling for $1.80 a piece (instead of the usual $2.00 each). The fixed costs to run the Chocolate Shoppe total $450 for the week, and he estimates that each chocolate costs about $0.60 to produce. Charlie estimates that he can produce up to 3,000 chocolates in one week. a) Write a function, C(n), to represent Charlie’s total costs for the week if he makes n chocolates. b) Write a function, R(n), to represent the revenue from the sale of n chocolates during the week before Valentine’s Day. c) Write a function, P(n), that represents Charlie’s profit from selling n chocolates during the week before Valentine’s Day. Show complete work to find the function.

d) Interpret the meaning of the statement P(300) = –90.

e) Determine the Practical Domain and Practical Range for P(n), then use that information to define an appropriate viewing window for the graph of P(n). Sketch the graph from your calculator in the space provided. Practical Domain:

Practical Range:

f) How many chocolates must Charlie sell in order to break even? Show complete work. Write your final answer as a complete sentence. Mark the break even point on the graph above.

Page 69

Lesson 2 – Functions and Function Operations

Lesson 2 Practice Problems Section 2.1: Combining Functions 1.

Let 𝑓(𝑥) = −3𝑥 + 2 and 𝑔(𝑥) = 𝑥 2 + 4𝑥 − 7. Find the following and simplify your result. a) 𝑓(4) + 𝑔(4) =

b) 𝑔(−3) − 𝑓(−3) =

c) 𝑓(2) ∙ 𝑔(2) =

d)

𝑔(0) 𝑓(0)

=

Page 70

Assessment

Lesson 2 – Functions and Function Operations

Assessment

2. Let 𝑓(𝑥) = 2𝑥 − 4 and 𝑔(𝑥) = 𝑥 2 − 9. Find the following. a) 𝑓(𝑥) + 𝑔(𝑥) =

b) 𝑔(𝑥) − 𝑓(𝑥) =

c) 𝑓(𝑥) ∙ 𝑔(𝑥) =

d)

𝑔(𝑥) 𝑓(𝑥)

=

3. Add, subtract and multiply the following functions. Simplify your answers. a) 𝑓(𝑥) = −4𝑥 + 7 and 𝑔(𝑥) = −3𝑥

𝑓(𝑥) + 𝑔(𝑥) =

𝑓(𝑥) − 𝑔(𝑥) =

𝑓(𝑥) ∙ 𝑔(𝑥) =

𝑔(𝑥) − 𝑓(𝑥) =

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Lesson 2 – Functions and Function Operations

Assessment

b) 𝑓(𝑥) = −𝑥 + 2 and 𝑔(𝑥) = −3𝑥 + 7 𝑓(𝑥) + 𝑔(𝑥) =

𝑓(𝑥) − 𝑔(𝑥) =

𝑓(𝑥) ∙ 𝑔(𝑥) =

𝑔(𝑥) − 𝑓(𝑥) =

c) 𝑓(𝑥) = 3𝑥 2 + 4𝑥 + 2 and 𝑔(𝑥) = 6𝑥 + 1 𝑓(𝑥) ∙ 𝑔(𝑥) =

𝑓(4) + 𝑔(−1) =

4. Simplify each of the following functions. Use only positive exponents in your final answer. a) 𝑓(𝑥) = 32𝑥 4 − 3𝑥 7 𝑎𝑛𝑑 𝑔(𝑥) = 6𝑥 4

f ( x) g ( x)

b) 𝑓(𝑥) = 48𝑥 9 − 16𝑥 3 + 4 𝑎𝑛𝑑 𝑔(𝑥) = −8𝑥 3 f (x) = g(x)

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Lesson 2 – Functions and Function Operations

Assessment

5. Use the tables of the functions below, find the following. x f(x)

−2 −3

−1 0

0 4

1 9

2 15

x g(x)

−2 12

−1 8

0 1

1 −3

2 −5

a) 𝑓(2) + 𝑔(2) = b) 𝑔(−1) − 𝑓(−1) = c) 𝑓(0) ∙ 𝑔(0) =

d)

𝑔(1) 𝑓(1)

=

6. Functions f(x) and g(x) are defined in the tables below. Use those tables to evaluate problems the problems below. x f(x)

-3 8

-2 6

0 3

1 2

4 5

5 8

8 11

10 15

12 20

x g(x)

0 1

2 3

3 5

4 10

5 4

8 2

9 0

11 -2

15 -5

a)

𝑓(5) =

b)

𝑔(5) =

c)

𝑓(5) + 𝑔(5) =

d)

𝑓(0) − 𝑔(0) =

e)

𝑓(8) ∙ 𝑔(8) =

f)

𝑓(4) ∙ 𝑔(0) =

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Lesson 2 – Functions and Function Operations

Assessment

7. Use the graph to determine each of the following. Assume integer answers. f(x)

g(x)

a) 𝑓(0) + 𝑔(0) =

b) 𝑓(1) − 𝑔(3) =

c) 𝑓(−2) ∙ 𝑔(5) =

d) 𝑓(−1) ∙ 𝑔(2) =

8. Functions f and g are defined below. Use those functions to evaluate the problems below. 𝑓 = {(−3,4), (−2,6), (−1,8), (0,6), (1, −2)}

𝑔 = {(−1,8), (0,2), (4,3), (8,4)}

a) 𝑓(−2) + 𝑔(0) =

b) 𝑓(1) − 𝑔(4) =

c) 𝑓(0) ∙ 𝑔(0) =

d) 𝑓(−1) ∙ 𝑔(8) =

Page 74

Lesson 2 – Functions and Function Operations

Assessment

Section 2.2: Applications of Function Operations 9. The function E(n) represents Ellen’s budgeted monthly expenses for the first half of the year 2013. In the table, n = 1 represents January 2013, n = 2 February 2013, and so on. n E(n)

1 2263

2 2480

3 2890

4 2263

5 2352

6 2550

The function L(n) shown in the table below represents Ellen’s monthly income for the first half of the year 2013. In the table, n = 1 represents January 2013, n = 2 February 2013, and so on. n L(n)

1 2850

2 2850

3 2850

4 2850

5 2850

6 2850

a) At the end of each month, Ellen puts any extra money into a savings account. The function S(n) represents the amount of money she puts into savings each month. Using the information above, complete the following table for the function S(n). n

1

2

3

4

5

6

S(n) b) Her goal is to save enough money to take a trip to Hawaii in July, 2013. She estimates that the trip will cost $2000. Will she be able to save up enough money to go to Hawaii in July? If so, how much extra money will he have to spend while she is there? If not, how much more does she need to earn?

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Lesson 2 – Functions and Function Operations

Assessment

10. Maria and Todd are organizing the 20 year reunion for their high school. The high school alumni association has given them $1000 for the event. They talk to the local caterer and find out the following: It will cost $15 per person plus a $50 setup fee to provide food for the event. It will cost $3 per person plus an $80 setup fee to rent the Meeting Hall at the local Holiday Motel. To help determine the costs, they come up with the following functions: The cost for food is $50 + $15 per person. F(x) = 15x + 50 The cost for the Hall is $80 + $3 per person H(x) = 3x + 80 In addition, they decide to charge each person $5 to get in the door. This can be modeled by the following function: Income for the event is $1000 from the alumni + $5 per person. I(x) = 5x + 1000 Given this information, answer the following questions. Show how you use the functions to calculate the answers. And give your final answers in complete sentences. If 400 people attend the event: a) How much will it cost for food?

b) How much will it cost to rent the Meeting Hall?

c) How much will it cost for food AND to rent the Meeting Hall? Show how you use the functions to calculate this. Hint: F(400) + H(400)

d) The final bill for the event is found by subtracting the costs from the income. What would the final bill for the event be?

e) Challenge question. How many people can attend if the costs have to equal the income?

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Lesson 2 – Functions and Function Operations

Assessment

11. Leonard has started a new business making cartoon bedspreads. His monthly expenses are $1322. Each bedspread costs $8.50 to produce. a) Complete the table below showing Leonard’s business costs as a function of the number of bedspreads he makes. n (number of bedspreads)

0

100

200

300

400

C(n) (Cost of n bedspreads) b) Leonard is selling each bedspread for $17.50. Complete the table below showing Leonard’s revenue as a function of the number of bedspreads he sells. n (number of bedspreads) R(n) (Revenue for n bedspreads)

0

100

200

300

400

c) The profit from Leonard’s business can be found by subtracting the cost function from the revenue function. Complete the table below showing Leonard’s profit as a function of the number of bedspreads he sells. n (number of bedspreads)

0

100

200

300

400

P(n) (Profit for n bedspreads) d) Using the information from parts a) through c), create algebraic functions for C, R and P. C(n) = R(n) = P(n) =

e) Using the table from part c), make a rough estimate for the number of bedspreads Leonard needs to sell for his business to break even. (Breaking even means profit = 0)

f) Using your formula for profit, P, determine the exact number of bedspreads Leonard needs to sell for his business to break even. (Breaking even means profit = 0)

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Lesson 2 – Functions and Function Operations

Assessment

Lesson 2 Assessment 1. If possible, simplify each of the following by combining like terms or using properties of exponents. a) 2n5 + 3n5 = _______________

b) 2n5 · 3n5 = _______________

c) 3n3 + 3n5 = _______________

d) 3n3 · 3n5 = _______________

2. The functions A and B are defined by the following tables x A(x)

–3 8

–2 6

0 3

1 2

4 5

5 8

8 11

10 15

12 20

x B(x)

0 1

2 3

3 5

4 10

5 4

8 2

9 0

11 –2

15 –5

Determine the values for each of the following. a) B(3)=_________

b) A(8)=_________

c) A(0)+B(0)=_______

d) A(8) – B(8)=_________

e) A(4) · B(4)=_________

f)

A(5) =________ B (5)

3. Let p(x) = x2 + 2x + 3 and r(x) = x – 5 . Determine each of the following. Show all work. Box your answers. a) p(x) – r(x)= b) p(0) · r(0)=

c) p(–2) + r(–2)=

d) r(7) – p(7)=

Page 78

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