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GPS Geometry Frameworks

Student Edition

Unit 6Inverse and Exponential Functions

GPS Geometry Unit 6 2nd Edition

2nd EditionOctober 27, 2009

Georgia Department of Education

Table of Contents

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

Unit Overview…………………………………………………...…………………4

Please Tell Me in Dollars and Cents Learning Task……………………………...7

Growing by Leaps and Bounds

Part 1……………………………………………………………………….15

Part 2……………………………………………………………………….16

Part 3……………………………………………………………………….17

Part 4……………………………………………………………………….19

Georgia Department of EducationKathy Cox, State Superintendent of Schools

October, 27 2009Copyright 2008 © All Rights Reserved

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GPS Geometry – Unit 6Inverse and Exponential Functions

Student Edition

INTRODUCTION : In GPS Algebra, 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 this unit, students apply their understanding of functions previously studied to explore the concept of inverse function. 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: 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:

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 .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

, including the use of restricted domains.



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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.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 first task focuses on exploration of inverse functions. In the first 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



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introducing the concept of inverse function. Students explore finding inverses from verbal statements, tables of values, algebraic formulas, and graphs.

The second 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.



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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.



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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:

.

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

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

,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: . What is the

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



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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, . What is the meaning of x and what is the meaning of f (x) when we use this notation?

c. Calculate . What is the meaning of this number?

d. Calculate , and simplify the result. What is the meaning of x and

what is the meaning of ?

e. Calculate using the formula from part d. Does your answer agree with your calculation from part c?

f. Calculate , 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 ?

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).1

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.

1 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/ .



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


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

.In Item 2, part b, we wrote this same formula by using the function f where 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 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 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?

e. Calculate the composite function , and simplify your answer. What is the meaning of x when we use x as input to this function?



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f. Calculate the composite function , 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 and 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 (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

for each input x in the domain of f,and

for each input x in the domain of h,

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

function h, and we can write f = .

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, , 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 inputAction of the function :

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

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

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



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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 and

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 . 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 ?

c. For any point (b, a) on the graph of , what is the corresponding point on the graph of f ? Justify your answer.



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x f (x)11

3 97

1015 3


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

if and only if .

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

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 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 , as in Item 2, so that is the temperature in degrees

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

that converts Celsius temperatures to Fahrenheit. Find a formula for .

c. Check that, for the functions g and from part b, and

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 with y and then solve for x in terms of y.



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

property, if , then . Use the formula for x, from part a, to write a

formula for in terms of y.

c. Write a formula for .

d. Find a formula for , where E is the function that converts Mexican pesos to Euros from Item 3, part b.

e. Find a formula for , 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, .

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 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 .

d. In general, what are the relationships between the domains and ranges of an invertible function and its inverse? Explain your reasoning.



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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 and

.

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

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

c. For functions in Item 6, part d, graph f, , 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 .

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?



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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.



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Day 0 1 2 3 4 5Number of people who know 1 2


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

, 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?



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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:

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?



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Age Birthday $ Amt from previous year plus Birthday

Total at year end

9 54 0 55.63831630101112

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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.



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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% 1662.7923 1662.79 600 11.25%24 700 11.50%25 800 11.75%

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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 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?



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4. Consider the function .

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.

b.

c.

d.

e.

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

shifts how? shifts how?

shifts how?



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