7.11: Inverse Functions
A planet's maximum distance from the sun (in astronomical units) is given by the formula
Guidance
By now, you are probably familiar with the term “inverse”. Multiplication and division are inverses of each other. More examples are addition and subtraction and the square and square root. We are going to extend this idea to functions. An inverse relation maps the output values to the input values to create another relation. In other words, we switch the
Example A
Find the inverse mapping of
Solution: Here, we will find the inverse of this relation by mapping it over the line
If we plot the two relations on the
The blue points are all the points in
If we were to fold the graph on
Domain of
Range of
Domain of
Range of
By looking at the domains and ranges of
Example B
Find the inverse of
Solution: This is a linear function. Let’s solve by doing a little investigation. First, draw the line along with
Notice the points on the function (blue line). Map these points over
The red line in the graph to the right is the inverse of
The equation of the inverse, read “
You may have noticed that the slopes of
Alternate Method: There is also an algebraic approach to finding the inverse of any function. Let’s repeat this example using algebra.
1. Change
2. Switch the
3. Solve for
The algebraic method will work for any type of function.
Example C
Determine if
Solution: There are two different ways to determine if two functions are inverses of each other. The first, is to find
Notice the
Therefore,
The inverse of
Alternate Method: The second, and easier, way to determine if two functions are inverses of each other is to use composition. If
Because
Intro Problem Revisit In the function
Switch the
3. Solve for
Now replace y and x with d and p. The inverse d is
Guided Practice
1. Find the inverse of
2. Find the inverse of
3. Determine if
Answers
1. Use the steps given in the Alternate Method for Example B.
2. Again, use the steps from Example B.
Yes,
3. First, find
Because
Explore More
Write the inverses of the following functions. State whether or not the inverse is a function.

(2,3),(−4,8),(−5,9),(1,1) 
(9,−6),(8,−5),(7,3),(4,3)
Find the inverses of the following functions algebraically. Note any restrictions to the domain of the inverse functions.

f(x)=6x−9 
f(x)=14x+3 
f(x)=x+7−−−−√ 
f(x)=x2+5 
f(x)=x3−11 
f(x)=x+16−−−−−√5
Determine whether

f(x)=23x−14 andg(x)=32x+21 
f(x)=x+58 andg(x)=8x+5 
f(x)=3x−7−−−−−√3 andg(x)=x33−7 
f(x)=xx−9,x≠9 andg(x)=9xx−1
Find the inverses of the following functions algebraically. Note any restrictions to the domain of the inverse functions. These problems are a little trickier as you will need to factor out the
Example:

x=3y+132y−11 First, switchx andy 
2xy−11x=3y+13 Multiply both sides by2y−11 to eliminate the fraction 
2xy−3y=11x+13 Now rearrange the terms to get both terms withy in them on one side and everything else on the other side 
y(2x−3)=11x+13 Factor out they 
y=11x+132x−3 Finally, Divide both sides by2x−3 to isolatey .
So, the inverse of

f(x)=x+7x,x≠0 
f(x)=xx−8,x≠8
Multistep problem
 In many countries, the temperature is measured in degrees Celsius. In the US we typically use degrees Fahrenheit. For travelers, it is helpful to be able to convert from one unit of measure to another. The following problem will help you learn to do this using an inverse function.
 The temperature at which water freezes will give us one point on a line in which
x represents the degrees in Celsius andy represents the degrees in Fahrenheit. Water freezes at 0 degrees Celsius and 32 degrees Fahrenheit so the first point is (0, 32). The temperature at which water boils gives us the second point (100, 212), because water boils at 100 degrees Celsius or 212 degrees Fahrenheit. Use this information to show that the equation to convert from Celsius to Fahrenheit isy=95x+32 orF=95C+32 .  Find the inverse of the equation above by solving for
C to derive a formula that will allow us to convert from Fahrenheit to Celsius.  Show that your inverse is correct by showing that the composition of the two functions simplifies to either
F orC (depending on which one you put into the other.)
 The temperature at which water freezes will give us one point on a line in which
11 function
A function is 11 if its inverse is also a function.composite function
A composite function is a function formed by using the output of one function as the input of another function . Composite functions are written in the form or .Function
A function is a relation where there is only one output for every input. In other words, for every value of , there is only one value for .Horizontal Line Test
The horizontal line test says that if a horizontal line drawn anywhere through the graph of a function intersects the function in more than one location, then the function is not onetoone and not invertible.inverse
Inverse functions are functions that 'undo' each other. Formally: and are inverse functions if .inverse function
Inverse functions are functions that 'undo' each other. Formally and are inverse functions if .Inverse Relation
An inverse relation is a relation with output values that are mapped to create input values for a new relation. The input values of the original relation would become the output values for the new relation.Onetoone
A function is onetoone if its inverse is also a function.Relation
A relation is any set of ordered pairs . A relation can have more than one output for a given input.Vertical Line Test
The vertical line test says that if a vertical line drawn anywhere through the graph of a relation intersects the relation in more than one location, then the relation is not a function.Image Attributions
Description
Learning Objectives
Here you'll learn how to find the inverse of a relation and function.
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Date Created:
Mar 12, 2013Last Modified:
Jun 04, 2015Vocabulary
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