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( f 1 f ) ( x ) = f 1 ( f ( x ) ) = f 1 ( y ) = x

This holds for all x in the domain of f . Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.

Given a function f ( x ) , we can verify whether some other function g ( x ) is the inverse of f ( x ) by checking whether either g ( f ( x ) ) = x or f ( g ( x ) ) = x is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)

For example, y = 4 x and y = 1 4 x are inverse functions.

( f 1 f ) ( x ) = f 1 ( 4 x ) = 1 4 ( 4 x ) = x

and

( f f 1 ) ( x ) = f ( 1 4 x ) = 4 ( 1 4 x ) = x

A few coordinate pairs from the graph of the function y = 4 x are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function y = 1 4 x are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

Inverse function

For any one-to-one function     f ( x ) = y , a function f 1 ( x ) is an inverse function    of f if f 1 ( y ) = x . This can also be written as f 1 ( f ( x ) ) = x for all x in the domain of f . It also follows that f ( f 1 ( x ) ) = x for all x in the domain of f 1 if f 1 is the inverse of f .

The notation f 1 is read f inverse.” Like any other function, we can use any variable name as the input for f 1 , so we will often write f 1 ( x ) , which we read as f inverse of x . Keep in mind that

f 1 ( x ) 1 f ( x )

and not all functions have inverses.

Identifying an inverse function for a given input-output pair

If for a particular one-to-one function f ( 2 ) = 4 and f ( 5 ) = 12 , what are the corresponding input and output values for the inverse function?

The inverse function reverses the input and output quantities, so if

f ( 2 ) = 4 ,  then  f −1 ( 4 ) = 2 ; f ( 5 ) = 12 ,  then f −1 ( 12 ) = 5.

Alternatively, if we want to name the inverse function g , then g ( 4 ) = 2 and g ( 12 ) = 5.

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Given that h 1 ( 6 ) = 2 , what are the corresponding input and output values of the original function h ?

h ( 2 ) = 6

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Given two functions f ( x ) and g ( x ) , test whether the functions are inverses of each other.

  1. Determine whether f ( g ( x ) ) = x or g ( f ( x ) ) = x .
  2. If either statement is true, then both are true, and g = f 1 and f = g 1 . If either statement is false, then both are false, and g f 1 and f g 1 .

Testing inverse relationships algebraically

If f ( x ) = 1 x + 2 and g ( x ) = 1 x 2 , is g = f 1 ?

g ( f ( x ) ) = 1 ( 1 x + 2 ) 2 = x + 2 2 = x

so

g = f 1  and  f = g 1

This is enough to answer yes to the question, but we can also verify the other formula.

f ( g ( x ) ) = 1 1 x 2 + 2 = 1 1 x = x
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If f ( x ) = x 3 4 and g ( x ) = x + 4 3 , is g = f 1 ?

Yes

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Determining inverse relationships for power functions

If f ( x ) = x 3 (the cube function) and g ( x ) = 1 3 x , is g = f 1 ?

f ( g ( x ) ) = x 3 27 x

No, the functions are not inverses.

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If f ( x ) = ( x 1 ) 3 and g ( x ) = x 3 + 1 , is g = f 1 ?

Yes

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Finding domain and range of inverse functions

The outputs of the function f are the inputs to f 1 , so the range of f is also the domain of f 1 . Likewise, because the inputs to f are the outputs of f 1 , the domain of f is the range of f 1 . We can visualize the situation as in [link] .

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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