Georgia Department of Education

GPS GeometryUnit 61st Teacher’s Edition

GPS Geometry Frameworks

Teacher’s Edition

Unit 6

Inverse, and Exponential Functions

1stEdition

March, 2011

GeorgiaDepartment of Education

Table of Contents

INTRODUCTION:

Notes on Please Tell Me in Dollar and Cents Learning Task

Notes on Growing by Leaps and Bounds Learning Task

GPS Geometry – Unit 6

Piecewise, Inverse, and Exponential Functions

Teacher’s 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 Unit 5 of GPS Algebra,studentsstudied general quadratic functions in depth. 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:

  • 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:

MM2A2. Students will explore exponential functions.

  1. Extend properties of exponents to include all integer exponents.
  2. 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.
  3. Graph functions as transformations of .
  4. Solve simple exponential equations and inequalities analytically, graphically, and by using appropriate technology.
  5. Understand and use basic exponential functions as models of real phenomena.
  6. Understand and recognize geometric sequences as exponential functions with domains that are whole numbers.
  7. Interpret the constant ratio in a geometric sequence as the base of the associated exponential function.

MM2A5. Students will explore inverses of functions.

  1. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range.
  2. Determine inverses of linear, quadratic, and power functions and functions of the form , including the use of restricted domains.
  3. Explore the graphs of functions and their inverses.
  4. 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 unit begins with tasks focus 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 introducing the concept of inverse function. Students explore finding inverses from verbal statements, tables of values, algebraic formulas, and graphs. In the second 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 next 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. The students will finish up the unit in applying exponential functions to geometric sequences.

TASKS:

The remaining content of this framework consists of student tasks or activities. The first task works with inverses and the second task concentrates on the introduction of exponential functions. Each activity isdesigned to allow students to build their own algebraic understanding through exploration. There is a student version, as well as a Teacher Edition version that includes notes for teachers and solutions.

Notes on Please Tell Me in Dollar and Cents Learning Task

This learning task addresses standard MM2A5 involving exploration of inverses of functions. The task introduces inverse functions through explorations of functions that convert from one quantity to another. Composition of functions is also introduced here so that students can use composition to verify that functions are inverses of each other (MM2A5, part d). Thus, the discussion of composition in GPS Geometry should be limited to that needed for the study of inverse functions. Students will investigate functions built through operations on functions, including composition, in GPS Pre-Calculus (MM4A4).

Students are introduced to composition of functions through conversion formulas for changing among Fahrenheit, Celsius, and Kelvin temperature scales and through currency conversions involving Japanese yen, Mexican pesos, Euros, and US dollar (Items 1 – 3). The familiar task of temperature conversion allows students to focus on the concept and meaning of function composition; currency conversions (the impetus for the title of the task) give students other examples to motivate the concept of function composition.

The need to convert temperatures, currency, and other quantities in the reverse direction is used to introduce the concept of inverse function. Students first see inverse functions through an example of a function that reverses, or undoes, the action of an original function (Item 4). Students are given a formal definition of inverse functions as functions whose compositions result in an identity function and then explore the meaning of the definition by starting from a verbal statement of the action of a function and producing a verbal statement of the action of its inverse to algebraic formulas. (Item 5) Students are asked to translate the verbal statements of functions and their inverses into algebraic formulas as an introduction to the idea of finding the formula for the inverse function (Item 6). Next, students apply the definition of inverse function to tables giving values for a function and its inverse to develop the Inverse Function Property (Items 7 – 8).

The Inverse Function Property is used as the guiding principle for an algebraic process to find a formula for the inverse given a formula for the original function. This algebraic process is developed by working with the inverse of the function that converts Fahrenheit to Celsius and then applied currency conversions and then to a function of the form which has a restricted domain (Items 9 – 11). In the later part of the task, students explore the relationship between the graph of an original function and its inverse function (Items 12 – 14).

Determining whether a given function has an inverse by considering whether it is one-to-one, finding inverses for quadratic and power functions, and future exploration of the characteristics of functions and their inverses are addressed in the next task.

Definitions and properties:

  • Composition of functions: 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.
  • Inverse functions: 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 = .
  • A function that has an inverse function is called invertible.
  • 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 . In terms of points on the graphs, (a, b) is a point on the graph of f if and only if (b, a) is a point on the graph of .

Supplies needed:

  • Calculator
  • Graphing utility
  • Graph paper and colored pencils

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.
  1. 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.

Comment(s):

Students are asked to do these calculations as introductory exploration of this function. Many students likely know the corresponding Celsius temperatures; this part asks them to use the formula to find the values.

Solution(s):

Freezing: , or freezing is 0°C.

Boiling: , or boiling is 100°C.

  1. 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.

Comment(s):

This part is about giving some sense of Kelvin temperatures and comprehending the meaning of the conversion formula.

Solution(s):

Freezing: , or freezing is 273°K.

Boiling: , or boiling is 373°K.

  1. Use the formulas from part a and part b to convert the following to °K: – 238°F, 5000°F .

Comment(s):

This part is designed to prepare students for idea of composing the two functions by having them do the two step process.

Solution(s):

Converting – 238°F to °K : ; Thus, – 238°F is 123°K.

Converting 5000°Fto °K :

Thus, 5000°F is 3303°K.

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.

  1. We now explore how the temperature conversions from Item 1, part c, provide an example of a composite function.
  1. 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?

Comment(s):

This part and the next involve the key conceptual step of moving to function notation where all of the inputs are expressed with the variable x. This is the notation they will use when they work with the inverse of a function.

Solution(s):

Here “x” is a temperature in degrees Fahrenheit and “g(x)” is the corresponding temperature in degrees Celsius.

  1. 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?

Here “x” is a temperature in degrees Celsius and “f(x)” is the corresponding temperature in degrees Kelvin.

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

Comment(s):

This part is designed to help students build familiarity with the notation. Students will organize their work in a variety of ways; the solution below provides one example.

Solution(s):

;

The value of is the temperature in °K that corresponds to 45°F.

  1. Calculate , and simplify the result. What is the meaning of x and what is the meaning of?

Comment(s):

This part is designed to introduce students to the types of calculations that they will need to do in verifying inverse functions using composition. Extensive practice in simplifying formulas created using composition is reserved for GPS Pre-Calculus, standard MM4A4, part c.

Solution(s):

Here “x” is a temperature in degrees Fahrenheit and “)” is the corresponding temperature in degrees Kelvin.

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

Comment(s):

This part insures that students realize that the simplified version of the composition formula eliminates the need for a two-step process and helps verify their calculations from part d.

Solution(s):

Yes, this computation gives the same answer.

  1. 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?

Comment(s):

This part is designed to make students notice the domain requirements for function composition. Here the issue is not whether the value can be calculated but instead is the issue of meaning. It brings home the point that, when we are working with functions that have real-world contexts, the contexts must “match-up” in order to form meaningful compositions.

Solution(s):

Here “x” is a temperature in degrees Celsius. We cannot associate a meaning to “)” relative to temperature conversion since “f(x)” is a temperature in degrees Kelvin, but an input to the function g should be a temperature in degrees Fahrenheit.

We now explore function composition further using the context of converting from one type of currency to another.

  1. 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]
  1. Using the rates above, write a function P such that P(x) is the number of Mexican pesos equivalent to x Japanese yen.

Comment(s):

Students may need to experiment with some specific values, such as starting with 1000 yen, in order to decide how to write the formula. Alternately, writing a proportion may help:

.

Writing conversion functions sometimes seems backwards to students. For example, they know that 1 foot = 12 inches, but, to write a function that converts from feet to inches, the formula would be I = 12F, where I is the number of inches and F is the number of feet. Thus, in this situation, function notation may help avoid confusion.

If students are bothered by the formula, it can also be explained with the unit factors method used extensively in science classes, as indicated in the solution below.

Solution(s):

; thus, , where x is a number of Japanese yen and P(x) is the corresponding number of Mexican pesos.

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

Comment(s):

This part is similar to part a, but here students must determine the meaning of x and E(x).

Solution(s):

, where x is a number of Mexican pesos and E(x) is the corresponding number of Euros.

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

, where x is a number of Euros and D(x) is the corresponding number of US dollars.

  1. 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.)

Comment(s):

This part and the two that follow are designed to reinforce student understanding of the notation for function composition as well as the concept. If students use current data from the internet and compare their numbers to the stated rate for conversion from yen to Euros, they may find differences in later digits because these values change over very short time intervals.

Unless students consistently round to the same number of significant digits, they will get different answers in part f below due to rounding error. If students do not follow rounding instructions and get different answers, there may need to be a review of scientific notation and the concept of significant digits. Scientific notation is included in the Grade 8 curriculum (standard M8N1, part j). This real-world application of currency conversion gives an opportunity to show that it is the consistent use of significant digits, and not the use of the same number of decimal places, that leads to reliable results.

Solution(s):

The function is .