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6.2 One-to-One Functions; Inverse Functions

In this section, we will study the following topics:

One-to-one functions

Finding inverse functions algebraically

Finding inverse functions graphically

Verifying that two functions are inverses, algebraically and graphically

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Inverse Functions

Example*:

x+3Let ( ) 2 - 3 and let g(x)= . Find and .

2f x x f g x g f x

Solution:

f xg g x2 3

xf fg x

3( )

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Inverse Functions

Notice, for each of the compositions f(g(x)) and g(f(x)), the input was x and the output was x.

That is because the functions f and g “undo” each other.

In this example, the functions f and g are __________ ___________.

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Definition of Inverse Functions

Definition of Inverse Function

f -1 is the inverse of function f

f(f -1(x)) = x for every x in the domain of f -1

AND

f -1(f(x))=x for every x in the domain of f

The domain of f must be equal to the range of f -1, and the range of f must be equal to the domain of f -1.

The notation used to show the inverse of function f is f -1 (read “f-inverse”).

We will use this very definition to verify that two functions are inverses.

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1 1( )

( )f x

f x

Watch out for confusing notation.

Always consider how it is used within the context of the problem.

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Use the definition of inverse functions to verify algebraically that

are inverse functions.

3 32

( ) 7 2 and ( )7

xf x x g x

Example

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One-to-one Functions

Not all functions have inverses. Therefore, the first step in any of these problems is to decide whether the function has an inverse.

A function f has an inverse if and only if f is ONE-TO-ONE.

A function is one-to-one if each y-value is assigned to only one x-value.

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One-to-one Functions

Using the graph, it is easy to tell if the function is one-to-one.

A function is one-to-one if its graph passes the H____________________ L_________ T__________ ;

i.e. each horizontal line intersects the graph at most ONCE.

one-to-one NOT one-to-one

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Use the graph to determine which of the functions are one-to-one.

1.

2.

3.

1( ) 4

3f x x

2( ) ( 3)g x x

( ) 3h x x

Example

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Finding the Inverse Function Graphically

To find the inverse GRAPHICALLY:

1. Make a table of values for function f.

2. Plot these points and sketch the graph of f

3. Make a table of values for f -1 by switching the x and y-coordinates of the ordered pair solutions for f

4. Plot these points and sketch the graph of f -1

The graphs of inverse functions f and f -1 are reflections of one another in the line y = x.

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Finding the Inverse Function Graphically (continued)

Example:

f(x)=3x-5

y=x

1( )f xx f(x)

0 -5

1 -2

2 1

3 4

x f-1(x)

-5 0

-2 1

1 2

4 3

Which of the following is the graph of the function below and its inverse?

A) B)

C) D)

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Finding the Inverse Function Algebraically

To find the inverse of a function ALGEBRAICALLY:

1. First, use the Horizontal Line Test to decide whether f has an inverse function. If f is not 1-1, do not bother with the next steps.

2. Replace f(x) with y.

3. Switch x and y

4. Solve the equation for y.

5. Replace y with f -1(x).

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Finding the Inverse Function Algebraically

Example

Find the inverse of each of the following functions, if possible.

1. 1

( ) 92

f x x

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Finding the Inverse Function Algebraically

Example

Find the inverse of each of the following functions, if possible.

2. ( ) 5f x x

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Finding the Inverse Function Algebraically

Example (cont.)

3. 3( ) 2 1f x x

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Finding the Inverse Function Algebraically

Example (cont.)

4. 2( ) 2; 0f x x x

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Java Applet

Click here to link to an applet demonstrating inverse functions.

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End of Section 6.2