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Mathematics II Frameworks

Student Edition

Unit 5 Piecewise, Inverse, and Exponential Functions

2nd Edition October 27, 2009

Georgia Department of Education

Mathematics II Unit 5 2nd Edition

Georgia Department of Education Kathy Cox, State Superintendent of Schools

October, 27 2009

Copyright 2008 © All Rights Reserved Unit 5: of 36

Table of Contents

Introduction ................................................................................................................ 3

Unit Overview…………………………………………………...…………………5

Definitions………………………………………………………………………….6

Planning a Race Strategy Learning Task…………………………………………..7

A Taxing Situation Learning Task………………………………………………..12

Parking Deck Pandemonium Learning Task…………………………………...…18

Please Tell Me in Dollars and Cents Learning Task……………………………...23

Growing by Leaps and Bounds

Part 1……………………………………………………………………….31

Part 2……………………………………………………………………….32

Part 3……………………………………………………………………….33

Part 4……………………………………………………………………….35



October, 27 2009


Mathematics II – Unit 5

Piecewise, Inverse, and Exponential Functions

Student Edition

INTRODUCTION:

In Mathematics I, students expanded their knowledge of functions to include basic quadratic,

cubic, absolute value, and rational functions. They learned to use the notation for functions and

to describe many important characteristics of functions. In Unit 1 of Mathematics II, students

studied general quadratic functions in depth. In this unit, students apply their understanding of

functions previously studied to analyze and construct piecewise functions and to explore the

concept of inverse function. The study of piecewise functions includes work with the greatest

integer and other step functions and an informal introduction to points of discontinuity. The

exploration of inverse functions leads to investigation of: the operation of function composition,

the concept of one-to-one function, and methods for finding inverses of previously studied

functions. The unit ends with an examination of exponential functions, equations, and

inequalities, with a focus on using basic exponential functions as models of real world

phenomena.

ENDURING UNDERSTANDINGS:

Functions with restricted domains can be combined to form a new function whose

domain is the union of the functions to be combined as long as the function values agree

for any input values at which the domains intersect.

Step functions are specific piecewise functions; some well-known step functions can be

defined using a single rule or correspondence.

One-to-one functions have inverse functions.

The inverse of a function is a function that reverses, or “undoes” the action of the original

function.

The graphs of a function and its inverse function are reflections across the line y = x.

Exponential functions can be used to model situations of growth, including the growth of

an investment through compound interest.

KEY STANDARDS ADDRESSED:

MM2A1. Students will investigate step and piecewise functions, including greatest integer

and absolute value functions.

b. Investigate and explain characteristics of a variety of piecewise functions including

domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of

discontinuity, intervals over which the function is constant, intervals of increase and

decrease, and rates of change.



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MM2A2. Students will explore exponential functions.

a. Extend properties of exponents to include all integer exponents.

b. Investigate and explain characteristics of exponential functions, including domain and

range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change,

and end behavior.

c. Graph functions as transformations of xf x a .

d. Solve simple exponential equations and inequalities analytically, graphically, and by

using appropriate technology.

e. Understand and use basic exponential functions as models of real phenomena.

MM2A5. Students will explore inverses of functions. a. Discuss the characteristics of functions and their inverses, including one-to-oneness,

domain, and range.

b. Determine inverses of linear, quadratic, and power functions and functions of the form

af x

x, including the use of restricted domains.

c. Explore the graphs of functions and their inverses.

d. Use composition to verify that functions are inverses of each other.

RELATED STANDARDS ADDRESSED:

MM2P1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

MM2P2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.

c. Develop and evaluate mathematical arguments and proofs.

d. Select and use various types of reasoning and methods of proof.

MM2P3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and

others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

MM2P4. Students will make connections among mathematical ideas and to other

disciplines. a. Recognize and use connections among mathematical ideas.



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b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

MM2P5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical

phenomena.

Unit Overview:

The unit begins with developing the idea of piecewise functions through a real-world context,

planning the strategy for running a race. Students write formulas for functions with restricted

domains, and learn to graph such functions with technology. They encounter piecewise

functions as functions created by creating the union of functions with disjoint domains.

The second task introduces another real world context, taxes. This task gives additional

exploration of piecewise functions and specifically introduces the concept of points of

discontinuity. The third task completes the tasks focused on piecewise functions and addresses

step functions, especially greatest integer, or floor, function and the least integer, or ceiling,

function.

The fourth and fifth tasks focus on exploration of inverse functions. In the fourth task of the

unit, conversions of temperatures among Fahrenheit, Celsius, and Kelvin scales and currency

conversions among yen, pesos, Euros, and US dollars provide a context for introducing the

concept of composition of functions. Reversing conversions is used as the context for

introducing the concept of inverse function. Students explore finding inverses from verbal

statements, tables of values, algebraic formulas, and graphs. In the fifth task, students explore

one-to-oneness as the property necessary for a function to have an inverse and see how

restricting the domain of a non-invertible function can create a related function that is invertible.

The last task introduces exponential functions and explores them through several applications to

situations of growth: the spread of a rumor, compound interest, and continuously compounded

interest. Students explore the graphs of exponential functions and apply transformations

involving reflections, stretches, and shifts.



October, 27 2009


Definitions:

Greatest integer function (floor function) – The greatest integer function is determined by

locating the greatest integer less than or equal to the x-value in question. Common notations:

xxf , xxf , or f x x .

Least integer function (ceiling function) – The least integer function is determined by locating

the least integer greater than or equal to the x-value in question. Notation: xxf .

Piecewise-defined function OR piecewise function – a function formed by the union of two or

more function rules, each with unique restricted domains.

Step function – a piecewise function whose graph consists of horizontal line segments that form

steps.

TASKS:

The remaining content of this framework consists of student tasks or activities. The first is

intended to launch the unit. Each activity is designed to allow students to build their own

algebraic understanding through exploration. The last task is a culminating task, designed to

assess student mastery of the unit. There is a student version, as well as a Teacher Edition

version that includes notes for teachers and solutions.



October, 27 2009


Planning a Race Strategy Learning Task

Saundra is a personal trainer at a local gym. Three of her clients asked her to help them train for

an upcoming 5K race. Although Saundra has no experience in training runners, she believed that

she could help each of her clients prepare to perform his or her best and agreed to develop plans

for them. Based on her knowledge of the physical condition of each client, Saundra developed

for each an individualized strategy to use in running the race and then designed a training plan to

support that race strategy.

1. One of the clients is Terrance, a very experienced runner. His plan is to run at a moderate

pace for the first two kilometers and then use his maximum speed for the final three

kilometers.

a. Based on data from Terrance’s workouts at the gym, Saundra has determined that

covering 2 kilometers in 10 minutes corresponds to a moderate pace for Terrance. Verify

that this pace corresponds to an average speed of 12 km/h (kilometers per hour), and then

express this pace in miles per hour.

b. Let f denote the function that expresses the distance covered after t minutes of running at

0.2 km/min for times 0 10t . Write the formula for f t , and graph the function f

by hand.

c. Saundra believes that, with training, Terrance can run fast enough to cover the last 3

kilometers of the race in 10 minutes. What average speed is this in kilometers per hour

and in miles per hour?

d. Let g denote the function that expresses the total distance covered after t minutes, for

times 10 20t , assuming that the distance covered is already 2 kilometers at time t =

10 and that running speed for the next 10 minutes is 0.3 km/min. Write the formula

for g t , and graph the function g.

e. The functions f and g together represent the proposed strategy for Terrance to use in

running the race. Draw the graphs of f and the g on the same coordinate axes. Consider

this as one graph What is the domain of the graph? Is it the graph of a function? Why or

why not?

2. Jim, the second client, is not as strong a runner as Terrance but he likes to start fast.

Saundra’s strategy for Jim to use in running the race is shown in the graph below.

a. Describe the domain for the function graphed above; state the numbers included and the

meaning of the numbers for Jim’s race strategy.

b. Describe the range for the function graphed above; state the numbers included and the

meaning of the numbers for Jim’s race strategy.

c. State any intercepts of the graph, and explain the meaning for Jim’s race strategy.



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d. Use the graph to write a verbal description of how Jim should run the 5K race. Your

description should note the points in the race at which he is supposed to change his

running pace and the proposed average speed during each segment.

The graphs in items 1 and 2 that show Terrance’s and Jim’s race strategies are examples of

functions that can not be specified using a single rule of correspondence between inputs and

outputs. These functions are constructed by combining two or more “pieces,” each of which is a

function with a restricted domain. The formal mathematical term for such a function is

piecewise function. The pieces in the example so far have come from the family of linear

functions, but the next example shows that the pieces can be selected from other function

families as well.

3. Sue is the least experienced runner among Saundra’s three clients. After working with Sue

for a few weeks, Saundra observed that Sue got her best time for long distances if she began

slowly and steadily increased her pace until she reached a speed that she could hold for a

while. However, Sue would always gradually slow down again near the end of the run.

Saundra decided to make a graph to illustrate how this pattern might work out in the 5K

race. She wanted to use her graphing utility to make the graph so that it would look

professional. We’ll follow Saundra’s steps to see how this is done.

a. Saundra decided to use the function 1f with the formula

1

21

128f t t for 0 16t to

model the distance Sue might cover during the first 16 minutes of the race. Knowing that

graphing utilities treat inequalities as logical statements which are either true, with output

value 1, or false, with output value 0, Saundra entered the function

1

2(1/128) / 0 ( 16)Y x x x

in her graphing software and obtained

the graph shown. If your graphing

utility provides for logical inequality

statements, use it to obtain a similar

graph with the same viewing window.

If it does not, state the graphing window

that is needed.

b. Explain why the function that Sue entered into her graphing utility graphs the function

1

21

128f t t just for the restricted domain 0 16t .

c. According to the model, what distance would Sue cover in the first 16 minutes? Explain

how you know.



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d. Using calculus, Saundra projected that Sue would be running at one-fourth of a kilometer

per minute by the time she had been running for 16 minutes. Saundra thought it

reasonable to assume that Sue would be able to continue running at this pace for the next

two kilometers. If so, how long would it take Sue to run two kilometers? How many

minutes into the race would Sue be after covering these next two kilometers?

e. Saundra entered the following

expression into her graphing software

and obtained the graph shown.

2 (1/ 4) 2 / 16 ( 24)Y x x x

If possible, use your graphing utility to

obtain a similar graph with the same

viewing window. If this is not possible

with your graphing utility, state the

viewing window.

In either case, explain how the expression leads to the graph shown.

f. Write the formula for the function, graphed in part e, for Saundra’s thoughts about how

Sue might run the second segment of her race. Use2f for the name of the function and t

for the input variable, and be sure to state the restriction on the domain.

g. Saundra decided to model the last segment of how Sue might run the race with the

function 3f defined by the formula

3

2132 5

64f t t . Using this model, when

would Sue finish the race?

h. What is the domain for the function 3f if this function represents the last segment of the

race? Explain. Make a hand-drawn sketch of the graph of the function 3f over this

domain, and then draw the graph using your graphing utility.

i. Draw the graphs of the functions 1f ,

2f , and 3f on the same coordinate axes to form the

graph of a single piecewise function that represents Saundra’s model of how Sue might

run the race. What feature of the graph demonstrates when Sue would speed up, slow

down, or maintain a steady pace? Explain.



October, 27 2009


When we combine two or more functions to form a piecewise function, we use notation that

indicates we have a single function. For a piecewise function f , we use the format:

f x

rule for function 1, domain for function 1

rule for function 2, domain for function 2

...

rule for last function, domain for last function

.

4. Write the piecewise function, f , that corresponds to Saundra’s model of how Sue might run

the 5K race.

As part of his training plan, Jim runs in his neighborhood most days of the week. The next three

items explore functions that model three different training runs for Jim. The inputs for each of

these functions represent time, as in the functions related to race strategies for Terrance, Jim, and

Sue. The outputs measure distances, but there is a difference from the previous items. Here the

distance is not necessarily the distance Jim has covered since he began running; instead the

distance is how far Jim is from his home.

5. The piecewise function d, defined below, represents one of Jim’s Saturday training runs and

models his distance from home, d(t), in miles as a function of time, t, in minutes since he left

home. 1

12

110

, 0 30

2.5, 30 35

6, 35 60

t t

d t t

t t

a. What is the domain of the function? What does the answer tell you about this particular

training run?

b. Draw a graph of the function d on graph paper and determine the range of the function.

c. What is the maximum value of the function? Interpret your answers in relation to Jim’s

training run.

d. What are the zeros of the function? Interpret these in relation to Jim’s training run.

e. Over what interval(s) is the function constant? Interpret your answer in relation to Jim’s

training run.

f. What are the intervals of increase and decrease for the function? Interpret your answers

in relation to Jim’s training run.



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6. The piecewise function k, defined below, represents one of Jim’s weekday training runs and

models his distance from home in kilometers as a function of time t since he left home.

2

2

2

0.01 1, 0 10

0.01 0.4 1, 10 30

0.01 0.8 17, 30 40

t t

k t t t t

t t t

a. Draw a graph of the function k on graph paper . You may find it helpful to use what you

know about transformations of the function 20.01y t .

b. State the domain and range, and interpret these for this training run.

c. Does the graph have any lines of symmetry? Explain.

d. What are the intercepts of the graph, if any? Interpret these for this training run.

e. What is the maximum value of the function? Interpret this value in relation to the

training run.

7. Consider a training run where Jim leaves from home, runs for half an hour so that his

distance from home is increasing at a constant rate of 4 miles per hour, and then for the next

half hour runs so that his distance from home is decreasing at 4 miles per hour.

a. Draw a graph of the function that models this training run.

b. Write a formula for the function as a piecewise function.

c. Write a single formula for this function using transformations of the absolute value

function.

8. Use the definition of absolute value of a real number to write a piecewise rule for the

absolute value function y = |x|.

9. Graph each of the following functions as a transformation of a function of the form

| |y a x , and then write a piecewise rule for it.

a. 2 | | 5f x x

b. | 3 |f x x

c. 12

| 2 | 2f x x

d. | 4 | 1f x x



October, 27 2009


A Taxing Situation Learning Task

1. Each year, the federal government publishes tables for employers to use to withhold income

tax for their employees. The amount of tax withheld depends on the employee’s marital

status, number of withholding allowances, and eligibility for Earned Income Credit, as well

as the length of the pay period. (Earned Income Credit, EIC, is an option for low wage

earners with children who meet a set of eligibility requirements.) There is a different table

for each combination of circumstances.

The table below lists the withholding rules for 2009 that apply to single persons paid on a

biweekly basis (who do not qualify for EIC).1

TABLE 1 (applies to biweekly salaries for employees who are single, no EIC):

Amount of Wages after subtracting withholding allowances

Income Tax to be Withheld

Not over $102 $0

Over But not over

$102 $400 10% of the amount over $102

$400 $1,362 $29.80 plus 15% of the amount over $400

$1,362 $3,242 $174.10 plus 25% of the amount over $1,362



$14,423 $4,162.08 plus 35% of the amount over $14,423

a. The number of dependents a person can claim gives an upper limit for the number of

withholding allowances. People who have income in addition to wages sometimes claim

fewer withholding allowances than the maximum in order to avoid paying extra taxes

when they file their income tax returns with the Internal Revenue Service (IRS).

For 2009, the US government amount for one withholding allowance on a biweekly

paycheck is $140.38; additional allowances are the same amount2. Find the taxes

withheld from a biweekly check for each of the following single US taxpayers in 2009.

Amounts are annual salaries.

(i) A firefighter who makes $46,380 (three withholding allowances)

(ii) A physician in family practice who makes $138,975 (no withholding allowance)

(iii) A medical laboratory technician who makes $28,200 (one withholding allowance)

(iv) A radiologist who makes $225,640 (one withholding allowance)

(v) A professional athlete who makes $1.5 million (no withholding allowance) and

has a contract that requires a biweekly paycheck

1 Table information obtained from Department of the Treasury, Internal Revenue Service, Notice 1036 (Rev.

November 2008), Catalog No. 21974B at http://www.irs.gov/pub/irs-pdf/n1036.pdf 2 Amount obtained from Department of the Treasury, Internal Revenue Service, Notice 1036 (Rev. November

2008), Catalog No. 21974B at http://www.irs.gov/pub/irs-pdf/n1036.pdf

http://www.irs.gov/pub/irs-pdf/n1036.pdf

http://www.irs.gov/pub/irs-pdf/n1036.pdf



October, 27 2009


(vi) A school bus driver who makes $16000 (two withholding allowances) and is paid

in biweekly checks spread throughout the year

b. Calculate the percentage of biweekly salary that is withheld for each taxpayer in part a,

and compare it to the corresponding percentage in the table.

c. Write the definition for a piecewise function, T, that can be used to calculate T x , the

income tax withheld, as a function of x, the wages after withholding allowances have

been subtracted, for taxpayers to whom TABLE 1 applies. Use the set of all nonnegative

real numbers as the domain for T.

d. Use the function T to recalculate the income tax withheld from biweekly paychecks for

each of the people in part a, and verify that you have the same values as before. If not,

correct the formula for T so that you have the same values.

2. In recent years, various individuals have made proposals for replacing the current system of

US income tax with alternate tax systems. The 2008 presidential candidate, John McCain,

supported one particular flat tax proposal. Each proposal for a flat tax involves a change in

what is to be taxed as well as a change in tax rates.

As a simple introduction to the idea of flat, in this item we consider a fictional proposal by a

character named Jacob Jones. Mr. Jones’ proposal would continue to tax income and would

withhold income tax from paychecks. He would keep the same system of withholding

allowances but raise the amount of salary exempt from income tax.

We provide TABLE 2 that indicates how Mr. Jones’ proposal would apply to taxpayers who

are single and paid on a biweekly basis. Thus, TABLE 2 applies to the same category of

taxpayers as TABLE 1.

TABLE 2 (would apply to biweekly salaries for employees who are single):


Income Tax to be Withheld

Not over $250 $0

Over $250 17% of the amount over $250

a. Write the definition for a function, F, that can be used to calculate F x , the income tax

withheld, as a function of x, the wages after withholding allowances have been

subtracted, for single taxpayers to whom TABLE 2 would apply. Use the set of all

nonnegative real numbers as the domain.

b. Compare the income tax that would be withheld from the biweekly paycheck under Mr.

Jones’ plan with the amount actually deducted using the 2009 tax schedule for each of the



October, 27 2009


six taxpayers from Item 1, part a. Who has more tax withheld and who has less under

Jones’ flat tax proposal?

The above comparison gives us a preliminary look at how the flat tax proposal differs from the

current US income tax. To get a full understanding of how the income tax schemes differ, we

would need to look at all taxpayers, whether single or married, over all of the possible of

payment periods from weekly to annually. To do so would take more time that we have to

devote to the topic, but we can get a fuller picture of the comparison between the two tax

schemes for taxpayers to whom Table 1 applies by examining the graphs of the functions T and

F.

3. Compare the functions T and F by graphing by doing the following.

a. Graph the function T by hand. Use a scale so that all pieces of the graph are indicated.

b. Graph the function F by hand on the same axes as the graph of T.

c. Graph T and F on the same axes using a graphing utility.

d. Describe where each graph is constant and state the constant amount.

e. Write sentences to interpret the answers to part d as a comparison between the two tax

schemes.

f. For each function, describe the rate of change over those intervals where the function is

not constant.

g. Write sentences to interpret the answers to part e as a comparison between the two tax

schemes.

h. To the nearest cent, for what biweekly salary (after subtracting withholding allowances),

is the amount of tax withheld the same in both schemes?

i. Write a paragraph summarizing the comparison for the two schemes for calculating the

amount of income tax withheld from the biweekly paycheck.

4. In Australia, the rules for calculating income tax use different percentages for different salary

ranges, as with the US system, but are based strictly on a percentage of income. Some call

such a system a progressive flat tax. Jill Jackson has a proposal for a progressive flat tax for

the United States. Her plan would result in the following rules for calculating withholding

for unmarried taxpayers.



October, 27 2009


TABLE 3 (would apply to biweekly salaries for employees who are single):


Income Tax to be

Withheld

Not over $300 $0

Over But not over

$300 $1300 15% of wages

$1300 $3000 17% of wages

$3000 $6200 20% of wages

$6200 $13500 22% of wages

$13500 25% of wages

a. Write the definition for a function, P, that can be used to calculate P x , the income tax

withheld, as a function of x, the wages after withholding allowances have been

subtracted, for single taxpayers to whom TABLE 3 would apply. Use the set of all

nonnegative real numbers as the domain for T.

b. Graph the function P by hand and using a graphing utility.

c. Let y represent a number in the range of P. What are the possible values for y?

d. How does the range of P differ from the ranges of the functions T and F?

The difference in the range of P and the ranges for the functions T and F shows up if we

consider drawing the graphs. For T and F, we can draw the graph over any interval of numbers

in the domain using one continuous motion without lifting the pencil, pen, or marker that we are

using to draw the graph. This is not the case with the graph of P. To draw the graph of the

function P, at certain points, we must interrupt our motion and lift our pencils (or pen or marker)

to get to the next section of the graph. The x-values at which we must lift our pencils before

continuing the graph are called points of discontinuity. A function, like T and F, for which the

domain is a single interval of real numbers and which has no points of discontinuity in that

interval is said to be continuous on its domain.



October, 27 2009




October, 27 2009


5. List the x-values that are points of discontinuity for the function P.

6. Consider the following three single taxpayers on a biweekly paycheck.

(i) A single taxpayer whose biweekly pay after withholding allowances are subtracted is

$298 and who then gets a raise that adds $10 to the biweekly pay.

(ii) A single taxpayer whose biweekly pay after withholding allowances are subtracted is


(iii) A single taxpayer whose biweekly pay after withholding allowances are subtracted is


a. How much of the $10 pay raise is included in the first taxpayer’s biweekly paycheck

under the three tax schemes modeled by the functions T, F, and P?

b. How much of the $25 pay raise is included in the second taxpayers biweekly paycheck


c. How much of the $700 pay raise is included in the third taxpayer’s biweekly paycheck


d. Which tax scheme allows a single taxpayer with a biweekly paycheck to keep the largest

part of a pay raise?



October, 27 2009


Parking Deck Pandemonium Learning Task

In this task, you will explore a particular type of piecewise function called a step function.

Although there are many different kinds of step functions, two common ones are the least

integer function, or the “ceiling function,” and the greatest integer function, sometimes called

the “floor function.”

The fee schedule at parking decks is often modeled using a step function. Let’s look at a few

different parking deck rates to see the step functions in action. (Most parking decks have a

maximum daily fee. However, for our exploration, we will assume that this maximum does not

exist.)

1. As you drive through town, Pete’s Parking Deck advertises free parking up to the first hour.

Then, the cost is $1 for each additional hour or part of an hour.

Thus, if you park at Pete’s Parking Deck for 59 minutes and 59 seconds, parking is free;

however, if the time shows at exactly 60 minutes, you pay $1. Similarly, if you park for any

time from 1 hour up to 2 hours, then you owe $1; but parking for exactly 2 hours costs $2.

a. Make a table listing some fees for parking at Pete’s for positive times that are 5 hours or

less. Be sure to include some non-integer values; write these in decimal form. Then draw

(by hand) the graph that illustrates the fee schedule at Pete’s for x hours, where

0 5x .

b. Use your graph to determine the fee if you park for 3 ½ hours. What about 3 hours, 55

minutes? 4 hours, 5 minutes?

c. What are the x- and y-intercepts of this graph? What is the interpretation in the context of

Pete’s Parking Deck?

d. What do you notice about the time, written in decimal form, and the corresponding fee?

Make a conjecture about the fee if you were to park at Pete’s Parking Deck for 10.5 hours

(assuming no maximum fee).

e. Write a piecewise function P to model the fee schedule at Pete’s Parking Deck.

2. If Pete’s Parking Deck allows fees to accumulate for multiple days for a car that is just left in

the lot, then, theoretically, there is no maximum fee. Thus, to write the rule for your

piecewise function model in Item 1, part d, immediately above, the statement of the rule for

P needed to show a pattern that continues forever.



October, 27 2009


There is a useful standard function that gives the same values as the function for the parking

fees at Pete’s Parking Deck but is defined for negative real numbers as well nonnegative

ones. This function is called the greatest integer function. The greatest integer function is

determined by locating the greatest integer that is less than or equal to the x-value in

question. For any real number x, x , is used to denote the greatest integer function applied

to x.

a. Evaluate each of the following by determining the greatest integer less than or equal to

the x-value, that is, let xxf , where x is any real number.

i) 3.6f ii) 0.4f iii) 0.4f iv) 1f v) 2.2f

b. Draw the graph of the greatest integer function, xxf , for the viewing

window 1010 x .

c. What is the domain of the greatest integer function, xxf ?

d. What is the range of the greatest integer function, xxf ?

e. What is the shape of the graph beyond the given viewing window? Can you indicate this

on your hand-drawn graph?

Several different notations are used for the greatest integer function. The two most common are

xxf , which we have used so far, and xxf . However, computer scientists use

another name for the greatest integer function; they call it the floor function, and use the

notation xxf . To help remember this notation, note that the bars on the brackets occur

only at the bottom (or floor) of the straight line segments.

The name “floor function” may be more helpful in remembering how the formula for the

function works. This function, by whatever name it is called, gives an integer value output that is

less than or equal to the value of the input number. Of course, there are many integers less than

or equal to any given number, so to make this a function, we choose the largest integer that meets

this condition. Choosing the largest integer that is less than or equal to the input number gives us

the name “greatest integer function” but that name can be misleading because the output is

always less than or equal to the input. The name “floor function” should help you remember

that the output is less than or equal to the input number just as the height of the floor of a room is

less than or equal to the height of any object in the room.



October, 27 2009


3. Practice working with the various notations for the greatest integer function.

a. For each expression below, rewrite the expression using each of the other notations.

i) 5.3 ii) 4.317 iii) 10.1 3.4 iv) 2.3 5.7 v) (1.34)( 6.8)

b. Evaluate each expression in part a.

4. Paula’s Parking Deck is down the street from Pete’s. Paula recently renovated her deck to

make the parking spaces larger, so she charges more per hour than Pete. Paula’s Parking

Deck offers free parking up to the first hour (i.e., the first 59 minutes). Then, the cost is $2

for each additional hour or part of an hour. (If you park for 1 ½ hours, you owe $2.)

a. Draw the graph that illustrates the fee schedule at Paula’s Parking Deck for x hours,

where 0 5x .

b. How does the graph for Paula’s Parking Deck compare with the graph of Pete’s Parking

Deck (from Item 1, part a)? To what graphical transformation does this change

correspond?

c. If you were to form a line by connecting the left endpoints of the steps in the graph for

Pete’s Parking Deck, found in answering Item 1, part a, what would be the equation of

the resulting linear function?

d. If you were to form a line by connecting the left endpoints of the steps in the graph for

Paula’s Parking Deck, found in answering part a for this item (Item 4), what would be the

equation of the resulting linear function?

e. How do your answers for parts c and d relate to your answer to part b in this item (Item

4)?

f. Write the function, g, in terms of the greatest integer function, that gives the same values

as the function for the parking fees at Paula’s Parking Deck but extends the domain to

include all real numbers.

g. Draw the graph of xgy over the domain 1010 x .

h. What are the domain and range for the function g?

5. Pablo’s Parking Deck is across the street from Paula’s deck. Pablo decided not to provide

any free parking. Pablo charges $1 for less than an hour, $2 for an hour or more but less than

two hours, and so forth, adding $1 whenever the time goes over the next hour mark. (If you

park for 59 minutes and 59 seconds, you owe $1; if you park for 1 hour, you owe $2; etc.)



October, 27 2009


a. Draw the graph that illustrates the fee schedule at Pablo’s Parking Deck for x hours,

where 0 5x .

b. How does the graph for Pablo’s Parking Deck compare with the graph for Pete’s Parking

Deck? To what graphical transformation does this change correspond?

c. Write the function, h, in terms of the greatest integer function, that gives the same values

as the function for the parking fees at Pablo’s Parking Deck but extends the domain to

include all real numbers. (What are the two different forms that this function could take?)

d. Draw the graph of xhy over the domain 1010 x .

e. What are the domain and range for the function h?

6. Padma’s Parking Deck is the last deck on the street. To be a bit more competitive, Padma

decided to offer parking for each full hour at $1/hour. (If you park for 59 minutes or exactly

1 hour, you owe $1; if you park for up to and including 2 hours, you owe $2.)

a. Draw the graph that illustrates the fee schedule at Padma’s Parking Deck for x hours,

where 0 5x .

b. To which of the graphs of the other parking deck rates is the graph for Padma’s Parking

Deck most similar? How are the graphs similar? How are they different?

The fee schedule at Padma’s Parking Deck is modeled by the least integer function, or ceiling

function. The least integer function is determined by locating the least integer that is greater

than or equal to the x-value in question. The least integer function is also called the ceiling

function and written with the following notation (analogous to the floor function notation):

c x x . To help remember this notation, note that the bars on the brackets occur only at

the top (or ceiling) of the straight line segments.

7. Let c x x .

a. Evaluate each of the following by determining the least integer greater than or equal to

the x-value.

i) 3.5c ii) 4c iii) 2.1c iv) .1c v) 1.6c

b. Draw the graph of y c x over the domain 1010 x .

c. What are the domain and range for the function c?

d. Suppose Padma chose to offer the first full hour free. After that, patrons would be

charged $1 for up through each full hour. What transformation of the least integer

function would model this parking fee structure?



October, 27 2009


8. As additional practice with step functions, graph each of the following.

For each function, state the parent function (either xxgxxf or ) and explain what

transformations have been applied to the parent function; state domain, range, and y-

intercept.

a. 2 1h x x b. 2xxj c. 2xxk



October, 27 2009


Please Tell Me in Dollars and Cents Learning Task

1. Aisha made a chart of the experimental data for her science project and showed it to her

science teacher. The teacher was complimentary of Aisha’s work but suggested that, for a

science project, it would be better to list the temperature data in degrees Celsius rather than

degrees Fahrenheit.

a. Aisha found the formula for converting from degrees Fahrenheit to degrees Celsius:

532

9C F .

Use this formula to convert freezing (32°F) and boiling (212°F) to degrees Celsius.

b. Later Aisha found a scientific journal article related to her project and planned to use

information from the article on her poster for the school science fair. The article included

temperature data in degrees Kelvin. Aisha talked to her science teacher again, and they

concluded that she should convert her temperature data again – this time to degrees

Kelvin. The formula for converting degrees Celsius to degrees Kelvin is

273K C .

Use this formula and the results of part a to express freezing and boiling in degrees

Kelvin.

c. Use the formulas from part a and part b to convert the following to °K: – 238°F,

5000°F .

In converting from degrees Fahrenheit to degrees Kelvin, you used two functions, the function

for converting from degrees Fahrenheit to degrees Celsius and the function for converting from

degrees Celsius to degrees Kelvin, and a procedure that is the key idea in an operation on

functions called composition of functions.

Composition of functions is defined as follows: If f and g are functions, the composite

function f g (read this notation as “f composed with g) is the function with the formula

( ) ( )f g x f g x ,

where x is in the domain of g and g(x) is in the domain of f.

2. We now explore how the temperature conversions from Item 1, part c, provide an example of

a composite function.

a. The definition of composition of functions indicates that we start with a value, x, and first

use this value as input to the function g. In our temperature conversion, we started with a

temperature in degrees Fahrenheit and used the formula to convert to degrees Celsius, so

the function g should convert from Fahrenheit to Celsius: 5

( ) 329

g x x . What is the

meaning of x and what is the meaning of g(x) when we use this notation?



October, 27 2009


b. In converting temperature from degrees Fahrenheit to degrees Kelvin, the second step is

converting a Celsius temperature to a Kelvin temperature. The function f should give us

this conversion; thus, ( ) 273f x x . What is the meaning of x and what is the meaning

of f (x) when we use this notation?

c. Calculate (45) (45)f g f g . What is the meaning of this number?

d. Calculate ( ) ( )f g x f g x , and simplify the result. What is the meaning of x and

what is the meaning of ( )f g x ?

e. Calculate (45) (45)f g f g using the formula from part d. Does your answer

agree with your calculation from part c?

f. Calculate ( ) ( )g f x g f x , and simplify the result. What is the meaning of x?

What meaning, if any, relative to temperature conversion can be associated with the value

of ( )g f x ?

We now explore function composition further using the context of converting from one type of

currency to another.

3. On the afternoon of May 3, 2009, each Japanese yen (JPY) was worth 0.138616 Mexican

pesos (MXN), each Mexican peso was worth 0.0547265 Euro (EUR), and each Euro was

worth 1.32615 US dollars (USD).3

a. Using the rates above, write a function P such that P(x) is the number of Mexican pesos

equivalent to x Japanese yen.

b. Using the rates above, write a function E that converts from Mexican pesos to Euros.

c. Using the rates above, write a function D that converts from Euros to US dollars.

3 Students may find it more interesting to look up current exchange values to use for this item and Item 9, which

depends on it. There are many websites that provide rates of exchange for currency. Note that these rates change

many times throughout the day, so it is impossible to do calculations with truly “current” exchange values. The

values in Item 3 were found using http://www.xe.com/ucc/ .

http://www.xe.com/ucc/



October, 27 2009


d. Using functions as needed from parts a – c above, what is the name of the composite

function that converts Japanese yen to Euros? Find a formula for this function. (Original

values have six significant digits; use six significant digits in the answer.)

e. Using functions as needed from parts a – c above, what is the name of the composite

function that converts Mexican pesos to US dollars? Find a formula for this function.

(Use six significant digits in the answer.)

f. Using functions as needed from parts a – c above, what is the name of the composite

function that converts Japanese yen to US dollars? Find a formula for this function. (Use

six significant digits in the answer.)

g. Use the appropriate function(s) from parts a - f to find the value, in US dollars, of the

following: 10,000 Japanese yen; 10,000 Mexican pesos; 10,000 Euros.

Returning to the story of Aisha and her science project: it turned out that Aisha’s project was

selected to compete at the science fair for the school district. However, the judges made one

suggestion – that Aisha express temperatures in degrees Celsius rather than degrees Kelvin. For

her project data, Aisha just returned to the values she had calculated when she first converted

from Fahrenheit to Celsius. However, she still needed to convert the temperatures in the

scientific journal article from Kevin to Celsius. The next item explores the formula for

converting from Kelvin back to Celsius.

4. Remember that the formula for converting from degrees Celsius to degrees Kelvin is

273K C .

In Item 2, part b, we wrote this same formula by using the function f where ( )f x represents

the Kelvin temperature corresponding to a temperature of x degrees Celsius.

a. Find a formula for C in terms of K, that is, give a conversion formula for going from °K

to °C.

b. Write a function h such that ( )h x is the Celsius temperature corresponding to a

temperature of x degrees Kelvin.

c. Explain in words the process for converting from degrees Celsius to degrees Kelvin. Do

the equation 273K C and the function f from Item 2, part b both express this idea?

d. Explain verbally the process for converting form degrees Kelvin to degrees Celsius. Do

your formula from part a above and your function h from part b both express this idea?



October, 27 2009


e. Calculate the composite function h f , and simplify your answer. What is the meaning

of x when we use x as input to this function?

f. Calculate the composite function f h , and simplify your answer. What is the meaning

of x when we use x as input to this function?

In working with the functions f and h in Item 4, when we start with an input number, apply one

function, and then use the output from the first function as the input to the other function, the

final output is the starting input number. Your calculations of h f and f h show that this

happens for any choice for the number x. Because of this special relationship between f and h ,

the function h is called the inverse of the function f and we use the notation 1

f (read this as

“f inverse”) as another name for the function h.

The precise definition for inverse functions is: If f and h are two functions such that

( ) ( )h f x h f x x for each input x in the domain of f,

and

( ) ( )f h x f h x x for each input x in the domain of h,

then h is the inverse of the function f, and we write h = 1f . Also, f is the inverse of the

function h, and we can write f = 1h .

Note that the notation for inverse functions looks like the notation for reciprocals, but in the

inverse function notation, the exponent of “–1 ” does not indicate a reciprocal.

5. Each of the following describes the action of a function f on any real number input. For each

part, describe in words the action of the inverse function, 1f , on any real number input.

Remember that the composite action of the two functions should get us back to the original

input.

a. Action of the function f : subtract ten from each input

Action of the function 1f :

b. Action of the function f : add two-thirds to each input


c. Action of the function f : multiply each input by one-half




October, 27 2009


d. Action of the function f : multiply each input by three-fifths and add eight


6. For each part of Item 5 above, write an algebraic rule for the function and then verify that the

rules give the correct inverse relationship by showing that 1 ( )f f x x and

1( )f f x x for any real number x.

Before proceeding any further, we need to point out that there are many functions that do not

have an inverse function. We’ll learn how to test functions to see if they have an inverse in the

next task. The remainder of this task focuses on functions that have inverses. A function that

has an inverse function is called invertible.

7. The tables below give selected values for a function f and its inverse function 1f .

a. Use the given values and the definition of inverse function to complete both tables.

b. For any point (a, b) on the graph of f, what is the corresponding point on the graph of 1f ?

c. For any point (b, a) on the graph of 1f , what is the corresponding point on the graph of

f ? Justify your answer.

x 1( )f x

3

5 10

7 6

3

11 1

x f (x)

11

3 9

7

10

15 3



October, 27 2009


As you have seen in working through Item 7, if f is an invertible function and a is the

input for function f that gives b as output, then b is the input to the function 1f that

gives a as output. Conversely, if f is an invertible function and b is the input to the

function 1f that gives a as output, then a is the input for function f that gives b as

output. Stated more formally with function notation we have the following property:

Inverse Function Property: For any invertible function f and any real numbers a

and b in the domain and range of f, respectively,

( )f a b if and only if 1f b a .

8. Explain why the Inverse Function Property holds, and express the idea in terms of points on

the graphs of f and 1f .

9. After Aisha had converted the temperatures in the scientific journal article from Kelvin to

Celsius, she decided, just for her own information, to calculate the corresponding Fahrenheit

temperature for each Celsius temperature.

a. Use the formula 5

329

C F to find a formula for converting temperatures in the

other direction, from a temperature in degrees Celsius to the corresponding temperature

in degrees Fahrenheit.

b. Now let 5

( ) 329

g x x , as in Item 2, so that ( )g x is the temperature in degrees

Celsius corresponding to a temperature of x degrees Fahrenheit. Then 1g is the function

that converts Celsius temperatures to Fahrenheit. Find a formula for 1( )g x .

c. Check that, for the functions g and 1g from part b, 1 ( )g g x x and 1( )g g x x

for any real number x.

Our next goal is to develop a general algebraic process for finding the formula for the inverse

function when we are given the formula for the original function. This process focuses on the

idea that we usually represent functions using x for inputs and y for outputs and applies the

inverse function property.

10. We now find inverses for some of the currency conversion functions of Item 3.

a. Return to the function P from Item 3, part a, that converts Japanese yen to Mexican

pesos. Rewrite the formula replacing ( )P x with y and then solve for x in terms of y.



October, 27 2009


b. The function 1P converts Mexican pesos back to Japanese yen. By the inverse function

property, if ( )y P x , then 1P y x . Use the formula for x, from part a, to write a

formula for 1P y in terms of y.

c. Write a formula for 1P x .

d. Find a formula for 1E x , where E is the function that converts Mexican pesos to Euros

from Item 3, part b.

e. Find a formula for 1D x , where D is the function that converts Euros to US dollars

from Item 3, part c.

11. Aisha plans to include several digital photos on her poster for the school-district science fair.

Her teacher gave her guidelines recommending an area of 2.25 square feet for photographs.

Based on the size of her tri-fold poster, the area of photographs can be at most 2.5 ft high.

Aisha thinks that the area should be at least 1.6 feet high to be in balance with the other items

on the poster.

a. Aisha needs to decide on the dimensions for the area for photographs in order to

complete her plans for poster layout. Define a function W such that W(x) gives the width,

in feet, of the photographic area when the height is x feet.

b. Write a definition for the inverse function, 1W .

In the remaining items you will explore the geometric interpretation of this relationship between

points on the graph of a function and its inverse.

12. We start the exploration with the function W from Item 11.

a. Use technology to graph the functions W and 1W on the same coordinate axes. Use a

square viewing window.

b. State the domain and range of the function W.

c. State the domain and range of the function 1W .

d. In general, what are the relationships between the domains and ranges of an invertible

function and its inverse? Explain your reasoning.



October, 27 2009


13. Explore the relationship between the graph of a function and the graph of its inverse function.

For each part below, use a standard, square graphing window with 10 10x and

10 10y .

a. For functions in Item 6, part a, graph f, 1f , and the line y = x on the same axes.

b. For functions in Item 6, part c, graph f, 1f , and the line y = x on the same axes.

c. For functions in Item 6, part d, graph f, 1f , and the line y = x on the same axes.

d. If the graphs were drawn on paper and the paper were folded along the line y = x, what

would happen?

e. Do you think that you would get the same result for the graph of any function f and its

inverse when they are drawn on the same axes using the same scale on both axes?

Explain your reasoning.

14. Consider the function 3

( )f xx

.

a. Find the inverse function algebraically.

b. Draw an accurate graph of the function f on graph paper and use the same scale on both

axes.

c. What happens when you fold the paper along the line y = x? Why does this happen?



October, 27 2009


Growing by Leaps and Bounds Learning Task

Part 1: Meet Linda

Linda’s lifelong dream had been to open her own business. After working and sacrificing and

saving, she finally had enough money to open up an ice cream business. The grand opening of

her business is scheduled for the Friday of Memorial Day weekend. She would like to have a

soft opening for her business on the Tuesday before. The soft opening should give her a good

idea of any supply or personnel issues she has and give her time to correct them before the big

official opening.

A soft opening means that the opening of the business is not officially announced; news of its

opening is just spread by word of mouth (see, not all rumors are bad!). Linda needs a good idea

of when she should begin the rumor in order for it to spread reasonably well before her soft

opening. She has been told that about 10% of the people who know about an event will actually

attend it. Based on this assumption, if she wants to have about 50 people visit her store on the

Tuesday of the soft opening, she will need 500 people to know about it.

1. Linda plans to tell one person each day and will ask that person to tell one other person each

day through the day of the opening, and so on. Assume that each new person who hears

about the soft opening is also asked to tell one other person each day through the day of the

opening and that each one starts the process of telling their friends on the day after he or she

first hears. When should Linda begin telling others about the soft opening in order to have at

least 500 people know about it by the day it occurs?

2. Let x represent the day number and let y be the number of people who know about the soft

opening on day x. Consider the day before Linda told anyone to be Day 0, so that Linda is

the only person who knows about the opening on Day 0. Day 1 is the first day that Linda

told someone else about the opening.

a. Complete the following table.

b. Graph the points from the table in part a.

3. Write an equation that describes the relationship between x (day) and y (number of people

who know) for the situation of spreading the news about the soft opening of Linda’s ice

cream store.

Day 0 1 2 3 4 5

Number of people who know 1 2



October, 27 2009


4. Does your equation describe the relationship between day and number who know about

Linda’s ice cream store soft opening completely? Why or why not?

Part 2: What if?

The spread of a rumor or the spread of a disease can be modeled by a type of function known as

exponential function; in particular, an exponential growth function. An exponential function

has the form xf x a b ,

where a is a non-zero real number and b is a positive real number other than 1. An exponential

growth function has a value of b that is greater than 1.

1. In the case of Linda’s ice cream store, what values of a and b yield an exponential function to

model the spread of the rumor of the soft store opening?

2. In this particular case, what is an appropriate domain for the exponential function? What

range corresponds to this domain?

3. In part 1, item 2, you drew a portion of the graph of this function. Does it make sense to

connect the dots on the graph? Why or why not?

4. How would the graph change if Linda had told two people each day rather than one and had

asked that each person also tell two other people each day?

5. How would the equation change if Linda had told two people each day rather than one and

had asked that each person also tell two other people each day? What would be the values of

a and b in this case?

6. How long would it take for at least 500 people to find out about the opening if the rumor

spread at this new rate?



October, 27 2009


Part 3: The Beginning of a Business

How in the world did Linda ever save enough to buy the franchise to an ice cream store? Her

mom used to say, “That Linda, why she could squeeze a quarter out of a nickel!” The truth is

that Linda learned early in life that patience with money is a good thing. When she was just

about 9 years old, she asked her dad if she could put her money in the bank. He took her to the

bank and she opened her very first savings account.

Each year until Linda was 16, she deposited her birthday money into her savings account. Her

grandparents (both sets) and her parents each gave her money for her birthday that was equal to

twice her age; so on her ninth birthday, she deposited $54 ($18 from each couple).

Linda’s bank paid her 3% interest, compounded quarterly. The bank calculated her interest using

the following standard formula:

1

ntr

A Pn

where A = final amount, P = principal amount, r = interest rate, n = number of times per year the

interest is compounded, and t is the number of years the money is left in the account.

1. Verify the first entry in the following chart, and then complete the chart to calculate how

much money Linda had on her 16th

birthday. Do not round answers until the end of the

computation, then give the final amount rounded to the nearest cent.

2. On her 16th

birthday, the budding entrepreneur asked her parents if she could invest in the

stock market. She studied the newspaper, talked to her economics teacher, researched a few

companies and finally settled on the stock she wanted. She invested all of her money in the

stock and promptly forgot about it. When she graduated from college on her 22nd

birthday,

she received a statement from her stocks and realized that her stock had appreciated an

average of 10% per year. How much was her stock worth on her 22nd

birthday?

Age Birthday $ Amt from previous

year plus Birthday

Total at year

end

9 54 0 55.63831630

10

11

12 .

.

.

.

.

.

.

.

.

.

.

.



October, 27 2009


3. When Linda graduated from college, she received an academic award that carried a $500

cash award. On her 22nd

birthday, she used the money to purchase additional stock. She

started her first job immediately after graduation and decided to save $50 each month. On

her 23rd

birthday she used the $600 (total of her monthly amount) savings to purchase new

stock. Each year thereafter she increased her the total of her savings by $100 and, on her

birthday each year, used her savings to purchase additional stock. Linda continued to learn

about stocks and managed her accounts carefully. On her 35th

birthday she looked back and

saw that her stock had appreciated at 11% during the first year after college and that the rate

of appreciation increased by 0.25% each year thereafter. At age 34, she cashed in enough

stock to make a down payment on a bank loan to purchase her business. What was her stock

worth on her 34th

birthday? Use a table like the one below to organize your calculations.

Age Amt from

previous year

Amt Linda

added from

savings that

year

Amount invested

for the year

Interest rate

for the year

Amt at

year end

22 998.01 500 1498.01 11.00% 1662.79

23 1662.79 600 11.25%

24 700 11.50%

25 800 11.75% .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.



October, 27 2009


Part 4: Some Important Questions

In learning about Linda’s journey, we have seen several examples of exponential growth

functions… the rumor, compounding interest in a savings account, appreciation of a stock. You

have already identified the exponential functions related to spreading the news of the soft

opening of Linda’s ice cream store. Now we’ll consider some other exponential functions you

have encountered in this task and explore some new ones.

1. The formula you used to find the value of Linda’s stocks on her 22nd

birthday (item 2 of Part

3) can be considered an application of an exponential function. Think of the values of P, r,

and n as constant and let the number of years vary so that the number of years is the

independent variable and the value of the stocks after t years is the dependent variable.

a. Write the equation for this exponential function.

b. What are the values of a and b so that it fits the definition of exponential function?

c. What point on the graph of this function did you find when you calculated the value of

Linda’s stock at age 22?

2. The formula you used to find the amount of money in Linda’s bank account when she was 10

years old can be considered an application of an exponential function where the number of

years, t¸ is the independent variable and the amount of money in the account at the end of t is

the dependent variable.

a. Write the equation for this exponential function.

b. What are the values of a and b so that it fits the definition of exponential function?

c. If Linda had not added money to the account each year, how much would she have had in

the account at age 16 from her original investment at age 9?

3. Consider the function 2xf x with an unrestricted domain.

a. Use a graphing utility to graph the function. In a future course you will learn the

meaning of the values of the function when x is not at integer.

b. What is the range of the function when the domain is all real numbers?

c. Why doesn’t the graph drop below the x-axis?



October, 27 2009


4. Consider the function 2 3xg x .

a. Predict how the graph of g is related to the graph of f from item 3 above.

b. Now use your graphing utility to graph the function g.

5. What is the range of the function g? How does this range compare to the range of the

function f? Explain why the ranges are related in this way.

6. The graph of an exponential function has a horizontal asymptote. Where is the asymptote

located in the graph of f? Where is the asymptote located in the graph of g?

7. Use your graphing utility to graph the following equations. Describe the graphs in parts b – e

as transformations of the graph of the function in part a.

a. 4xf x

b. 2 4xf x

c. 34xf x

d. 4 2xf x

e. 4xf x

8. Make some generalizations. What impact did each of the changes you made to the equation

have on the graph?

shifts how?

shifts how? 14 5xf x

shifts how?

unit 5 piecewise, inverse, and exponential functionsuat.georgiastandards.org/frameworks/gso...

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