Mathematics II Frameworks Student Edition Unit 5 Piecewise, Inverse, and Exponential Functions

Mathematics II Frameworks Student Edition Unit 5 Piecewise, Inverse, and Exponential Functions 2 nd Edition October 27, 2009

Table of Contents Introduction 3 Unit Overview 5 Definitions 6 Planning a Race Strategy Learning Task 7 A Taxing Situation Learning Task 2 Parking Deck Pandemonium Learning Task 8 Please Tell Me in Dollars and Cents Learning Task 23 Growing by Leaps and Bounds Part 3 Part 2 32 Part 3 33 Part 4 35 Unit 5: Page 2 of 36

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 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: MM2A 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 Unit 5: Page 3 of 36

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 x f 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 a f x, including the use of restricted domains x c Explore the graphs of functions and their inverses d Use composition to verify that functions are inverses of each other RELATED STANDARDS ADDRESSED: MM2P 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 Unit 5: Page 4 of 36

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 Unit 5: Page 5 of 36

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: f x x, f x x, 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: f x x 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 Unit 5: Page 6 of 36

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 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 0 minutes corresponds to a moderate pace for Terrance Verify that this pace corresponds to an average speed of 2 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 02 km/min for times 0 t 0 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 0 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 0 t 20, assuming that the distance covered is already 2 kilometers at time t = 0 and that running speed for the next 0 minutes is 03 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 Unit 5: Page 7 of 36

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 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 2 a Saundra decided to use the function fwith the formula f t t for 0 t 6 to 28 model the distance Sue might cover during the first 6 minutes of the race Knowing that graphing utilities treat inequalities as logical statements which are either true, with output value, or false, with output value 0, Saundra entered the function 2 Y (/28) x / 0 x ( x 6) 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 2 f t t just for the restricted domain 0 t 6 28 c According to the model, what distance would Sue cover in the first 6 minutes? Explain how you know Unit 5: Page 8 of 36

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 6 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 Y2 (/ 4) x 2 / 6 x ( x 24) 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 Use f 2 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 f 3 defined by the formula f 3 t 2 t 32 64 5 Using this model, when would Sue finish the race? h What is the domain for the function f 3 if this function represents the last segment of the race? Explain Make a hand-drawn sketch of the graph of the function f 3 over this domain, and then draw the graph using your graphing utility i Draw the graphs of the functions f, 2 f, and 3 f 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 Unit 5: Page 9 of 36

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, domain for function 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 t, 0 t 30 d t 2 25, 30 t 35 0 t 6, 35 t 60 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 Unit 5: Page 0 of 36

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 00t, 0 t 0 k t t t t 2 00 04, 0 30 2 00t 08t 7, 30 t 40 a Draw a graph of the function k on graph paper You may find it helpful to use what you 2 know about transformations of the function y 00t 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 f x 2 x 5 b f x x 3 c f x 2 x 2 2 d f x x 4 Unit 5: Page of 36

A Taxing Situation Learning Task 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) TABLE (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 $02 $0 Over But not over $02 $400 0% of the amount over $02 $400 $,362 $2980 plus 5% of the amount over $400 $,362 $3,242 $740 plus 25% of the amount over $,362 $3,242 $6,677 $6440 plus 28% of the amount over $3,242 $6,677 $4,423 $60590 plus 33% of the amount over $6,677 $4,423 $4,6208 plus 35% of the amount over $4,423 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 $4038; additional allowances are the same amount 2 Find the taxes withheld from a biweekly check for each of the following single US taxpayers in 2009 Amounts are annual salaries (i) (ii) (iii) (iv) (v) A firefighter who makes $46,380 (three withholding allowances) A physician in family practice who makes $38,975 (no withholding allowance) A medical laboratory technician who makes $28,200 (one withholding allowance) A radiologist who makes $225,640 (one withholding allowance) A professional athlete who makes $5 million (no withholding allowance) and has a contract that requires a biweekly paycheck Table information obtained from Department of the Treasury, Internal Revenue Service, Notice 036 (Rev November 2008), Catalog No 2974B at http://wwwirsgov/pub/irs-pdf/n036pdf 2 Amount obtained from Department of the Treasury, Internal Revenue Service, Notice 036 (Rev November 2008), Catalog No 2974B at http://wwwirsgov/pub/irs-pdf/n036pdf Unit 5: Page 2 of 36

(vi) A school bus driver who makes $6000 (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 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 TABLE 2 (would apply to biweekly salaries for employees who are single): Amount of Wages after subtracting withholding allowances Income Tax to be Withheld Not over $250 $0 Over $250 7% 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 Unit 5: Page 3 of 36

six taxpayers from Item, 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 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 Unit 5: Page 4 of 36

TABLE 3 (would apply to biweekly salaries for employees who are single): Amount of Wages Income Tax to be after subtracting withholding allowances Withheld Not over $300 $0 Over But not over $300 $300 5% of wages $300 $3000 7% of wages $3000 $6200 20% of wages $6200 $3500 22% of wages $3500 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 Unit 5: Page 5 of 36

Unit 5: Page 6 of 36

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 $0 to the biweekly pay (ii) A single taxpayer whose biweekly pay after withholding allowances are subtracted is $383 and who then gets a raise that adds $25 to the biweekly pay (iii) A single taxpayer whose biweekly pay after withholding allowances are subtracted is $5996 and who then gets a raise that adds $700 to the biweekly pay a How much of the $0 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 under the three tax schemes modeled by the functions T, F, and P? c How much of the $700 pay raise is included in the third taxpayer s biweekly paycheck under the three tax schemes modeled by the functions T, F, and P? d Which tax scheme allows a single taxpayer with a biweekly paycheck to keep the largest part of a pay raise? Unit 5: Page 7 of 36

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) As you drive through town, Pete s Parking Deck advertises free parking up to the first hour Then, the cost is $ 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 $ Similarly, if you park for any time from hour up to 2 hours, then you owe $; 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 x 5 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 05 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, part d, immediately above, the statement of the rule for P needed to show a pattern that continues forever Unit 5: Page 8 of 36

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 f x x, where x is any real number i) f 36 ii) f 04 iii) f 04 iv) f v) f 22 b Draw the graph of the greatest integer function, f x x window 0 x 0, for the viewing c What is the domain of the greatest integer function, f x x? d What is the range of the greatest integer function, f x x? 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 f x x, which we have used so far, and f x x However, computer scientists use another name for the greatest integer function; they call it the floor function, and use the notation f x x 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 Unit 5: Page 9 of 36

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) 53 ii) 437 iii) 0 34 iv) 23 57 v) (34)( 68) 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 (ie, the first 59 minutes) Then, the cost is $2 for each additional hour or part of an hour (If you park for ½ hours, you owe $2) a Draw the graph that illustrates the fee schedule at Paula s Parking Deck for x hours, where 0 x 5 b How does the graph for Paula s Parking Deck compare with the graph of Pete s Parking Deck (from Item, 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, 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 y g x over the domain 0 x 0 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 $ for less than an hour, $2 for an hour or more but less than two hours, and so forth, adding $ whenever the time goes over the next hour mark (If you park for 59 minutes and 59 seconds, you owe $; if you park for hour, you owe $2; etc) Unit 5: Page 20 of 36

a Draw the graph that illustrates the fee schedule at Pablo s Parking Deck for x hours, where 0 x 5 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 y h x over the domain 0 x 0 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 $/hour (If you park for 59 minutes or exactly hour, you owe $; 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 x 5 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) c 35 ii) c 4 iii) c 2 iv) c v) c 6 b Draw the graph of y c x over the domain 0 x 0 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 $ for up through each full hour What transformation of the least integer function would model this parking fee structure? Unit 5: Page 2 of 36

8 As additional practice with step functions, graph each of the following For each function, state the parent function (either f x x or g x x ) and explain what transformations have been applied to the parent function; state domain, range, and y- intercept a h x 2 x b j x x 2 c k x x 2 Unit 5: Page 22 of 36

Please Tell Me in Dollars and Cents Learning Task 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: 5 C F 32 9 Use this formula to convert freezing (32 F) and boiling (22 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 K C 273 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, 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 5 the function g should convert from Fahrenheit to Celsius: g( x) x 32 What is the 9 meaning of x and what is the meaning of g(x) when we use this notation? Unit 5: Page 23 of 36

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, f ( x) x 273 What is the meaning of x and what is the meaning of f (x) when we use this notation? c Calculate f g (45) f g (45) 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 f g (45) f g (45) 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 03866 Mexican pesos (MXN), each Mexican peso was worth 00547265 Euro (EUR), and each Euro was worth 3265 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://wwwxecom/ucc/ Unit 5: Page 24 of 36

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: 0,000 Japanese yen; 0,000 Mexican pesos; 0,000 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 K C 273 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 hx () 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 K C 273 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? Unit 5: Page 25 of 36

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 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 and h f ( x) h f ( x) x for each input x in the domain of f, 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 = function h, and we can write f = h f Also, f is the inverse of the Note that the notation for inverse functions looks like the notation for reciprocals, but in the inverse function notation, the exponent of 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, f, 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 f : b Action of the function f : add two-thirds to each input Action of the function f : c Action of the function f : multiply each input by one-half Action of the function f : Unit 5: Page 26 of 36

d Action of the function f : multiply each input by three-fifths and add eight Action of the function f : 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 f f ( x) x for any real number x f f ( x) x and 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 f a Use the given values and the definition of inverse function to complete both tables x f (x) 3 9 7 0 5 3 x f () x 3 5 0 7 6 3 b For any point (a, b) on the graph of f, what is the corresponding point on the graph of f? c For any point (b, a) on the graph of f? Justify your answer f, what is the corresponding point on the graph of Unit 5: Page 27 of 36

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 f that gives a as output Conversely, if f is an invertible function and b is the input to the function f 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 f b a 8 Explain why the Inverse Function Property holds, and express the idea in terms of points on the graphs of f and f 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 5 a Use the formula C F 32 to find a formula for converting temperatures in the 9 other direction, from a temperature in degrees Celsius to the corresponding temperature in degrees Fahrenheit 5 b Now let g( x) x 32, as in Item 2, so that gx ( ) is the temperature in degrees 9 Celsius corresponding to a temperature of x degrees Fahrenheit Then g is the function that converts Celsius temperatures to Fahrenheit Find a formula for g ( x ) c Check that, for the functions g and for any real number x g from part b, g g( x) x and g g ( x) 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 0 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 Px ( ) with y and then solve for x in terms of y Unit 5: Page 28 of 36

b The function P converts Mexican pesos back to Japanese yen By the inverse function property, if y P() x, then P y x Use the formula for x, from part a, to write a formula for c Write a formula for P y in terms of y P x d Find a formula for E from Item 3, part b e Find a formula for D from Item 3, part c x, where E is the function that converts Mexican pesos to Euros x, where D is the function that converts Euros to US dollars 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 225 square feet for photographs Based on the size of her tri-fold poster, the area of photographs can be at most 25 ft high Aisha thinks that the area should be at least 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, W In the remaining items you will explore the geometric interpretation of this relationship between points on the graph of a function and its inverse 2 We start the exploration with the function W from Item a Use technology to graph the functions W and W 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 W d In general, what are the relationships between the domains and ranges of an invertible function and its inverse? Explain your reasoning Unit 5: Page 29 of 36

3 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 0 x 0 and 0 y 0 a For functions in Item 6, part a, graph f, b For functions in Item 6, part c, graph f, c For functions in Item 6, part d, graph f, f f f, and the line y = x on the same axes, and the line y = x on the same axes, 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 3 4 Consider the function f( x) x 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? Unit 5: Page 30 of 36

Growing by Leaps and Bounds Learning Task Part : 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 0% 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 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 is the first day that Linda told someone else about the opening a Complete the following table Day 0 2 3 4 5 Number of people who know 2 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 Unit 5: Page 3 of 36

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 x f x a b, where a is a non-zero real number and b is a positive real number other than An exponential growth function has a value of b that is greater than 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, 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? Unit 5: Page 32 of 36

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 6, 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 ($8 from each couple) Linda s bank paid her 3% interest, compounded quarterly The bank calculated her interest using the following standard formula: nt r A P n 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 Verify the first entry in the following chart, and then complete the chart to calculate how much money Linda had on her 6 th birthday Do not round answers until the end of the computation, then give the final amount rounded to the nearest cent Age Birthday $ Amt from previous year plus Birthday Total at year end 9 54 0 556383630 0 2 2 On her 6 th 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 22 nd birthday, she received a statement from her stocks and realized that her stock had appreciated an average of 0% per year How much was her stock worth on her 22 nd birthday? Unit 5: Page 33 of 36

3 When Linda graduated from college, she received an academic award that carried a $500 cash award On her 22 nd 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 23 rd 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 $00 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 35 th birthday she looked back and saw that her stock had appreciated at % during the first year after college and that the rate of appreciation increased by 025% 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 34 th 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 9980 500 4980 00% 66279 23 66279 600 25% 24 700 50% 25 800 75% Unit 5: Page 34 of 36

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 The formula you used to find the value of Linda s stocks on her 22 nd 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 0 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 6 from her original investment at age 9? 3 Consider the function 2 x f 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? Unit 5: Page 35 of 36

4 Consider the function g x 2 x 3 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 f x 4 x b f x 2 4 x c f x 4 x 3 d f x 4 x 2 e f x 4 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? f x 4 x 5 shifts how? Unit 5: Page 36 of 36