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Inverse Functions

Site: Clare-Gladwin RESD
Course: Michigan Algebra II KHauck
Book: Inverse Functions
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Date: Sunday, May 19, 2024, 8:25 PM

Description

Table of contents

Introduction

An inverse function is a function that undoes another function, much like addition undoes subtraction. Inverse functions use inverse operations to turn inputs into outputs and vice versa.

If a function answers the question: “Alice worked this long; how much money has she made?” Then its inverse answers the question: “Alice made this much money; how long did she work?"

If a function answers the question: “How many hours of music fit on 12 CDs?” Then its inverse answers the question: “How many CDs do you need for 3 hours of music?”

If a function is modeled by the rule, inverse function 1 then the inverse function is modeled by inverse2y . Since multiplication and division are inverse operations, they will create inverse functions.

A common example of an inverse function is the Celsius-to-Fahrenheit conversion:

CtoFconversion

where C is the Celsius temperature and F the Fahrenheit temperature.

Substituting the input 100°C into the first function gives an output of 212°F. Inversely, substituting the input 212°F into the second function gives an output of 100°C.

Notation

The function, f -1 (x) , is defined as the inverse function of f(x) if it consistently reverses the process of f(x). That is, if f(x) turns a into b, then f -1 (x) must turn b into a. f -1 (x) is not an exponent and is read “the inverse of function f.” It is meant to be an inverse function, not a reciprocal.


Composition

More concisely and formally, f -1(x) is the inverse function of f(x) if and only if f (f -1(x)) = x and f -1(f(x)) = x .

Example 1
Are compInversEx1-1 and compFuncEx1-2 inverses?

Step 1. Find the composition f (g(x)) Step 2. Find the composition g(f(x))
step 1
 Step 2

Step 3. Determine if the functions are inverses.

Since f(g(x)) = x and g(f(x)) = x, the functions are inverses.




Example 2

Are InverseEx2-1 and InverseEx2-2 inverses?


Step 1. Find the composition f(g(x)) Step 2. Find the composition g(f(x))
InverseEx2-3
 InverseEx2-4











Step 3. Determine if the functions are inverses.

Since InverseEx2-5 and InverseEx2-6 , the functions are NOT
inverses. To be inverses, both compositions must produce only x.



Inverse Function Graph

The graphs of inverse functions reflect over the line y = x. See the graph below:

InverseFunctionGraph


*Note: the reflection of the point (a,b) over the line y = x is the point (b,a) .

Example 1

Let f(x) = x3 + 2. Then f(2) = 10 and the point (2,10) is on the graph of f. The inverse of f must take 10 back to 2, i.e. f -1(10) = 2, so the point (10,2) is on the graph of f -1. The point (10,2) is the reflection over the line y = x of the point (2,10). The same argument can be made for all points on the graphs of f and f -1 . Therefore, the functions are inverses.

The graph of f -1 is the reflection over the line y = x of the graph of f.

graphf-1



Example 2

If f has an inverse, then its graph will be the reflection of the graph of f over the line y = x. The graph of f and its reflection over y = x are drawn below.

inversegrapEX2

*Note: The reflected graph does not pass the vertical line test, so it is not the graph of a function.
Therefore, the function y = x2 does not have an inverse function.

The reflection of the graph is an inverse relation, but it is not a function.



Video Lesson

To learn how to graph inverse functions and relations, select the following link:

Graphing Inverse Relations

 

Finding Inverse Functions

Remember that an inverse function reverses the inputs and outputs. When we graph functions, we always represent the incoming number as x and the outgoing number as y. To find the inverse function switch the x and y values and then solve for y.

Example: Find the inverse function of findInverseFunct1 .

Step 1. Replace f(x) with y in the equation describing the function.

findInverseFunc2

Step 2. Interchange x and y. In other words, replace every x with a y and vice versa.

findInversFunc3


continued...

Example continued

Step 3. Solve for y.

findInverseFunc4

Step 4. Replace y with f-1(x).

findInverseFunc5

Step 5. Test to make sure this solution satisfies the definition of an inverse function.

Pick a number, and plug it into the original function: f(9) = 3

See if the inverse function reverses this process: f-1(3) = 9

Video Lessons

To learn how to write inverse functions, select the following link:

Writing Inverse Functions

To learn how to write multi-step inverse functions, select the following link:

Writng Multi-Step Inverse Functions

Interactive Activity

The graphs of a function and inverses of a function.

Inverses Activity


Practice

Familes of Functions Inverses Worksheet

*Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,” refresh your browser.


Video Lesson

Answer Key

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Sources

Sources used in this book:

Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project

Holt, Rinehart & Winston, . "Exponential and Logarithmic Functions." http://my.hrw.com/math06_07/nsmedia/homework_help/alg2/alg2_ch07_02_homeworkhelp.html (accessed 06/30/2010).

Kenny Felder, "Function Concepts -- Inverse Functions," Connexions, December 30, 2008, http://cnx.org/content/m18198/1.2/.

Ryson & Blond, " Graphs of a Function and the Inverse of the Function." http://www.members.shaw.ca/ron.blond/TLE/InvFcn.APPLET/index.html (accessed 06/30/2010).

UNC Wilmington, "Inverse Functions." http://www.uncwil.edu/courses/mat111hb/functions/inverse/inverse.html (accessed 6/29/2010).