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Draw graphs of the functions f   and   f 1 from [link] .

Graph of f(x) and f^(-1)(x).
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Is there any function that is equal to its own inverse?

Yes. If f = f 1 , then f ( f ( x ) ) = x , and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because

1 1 x = x

Any function f ( x ) = c x , where c is a constant, is also equal to its own inverse.

Access these online resources for additional instruction and practice with inverse functions.

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Key concepts

  • If g ( x ) is the inverse of f ( x ) , then g ( f ( x ) ) = f ( g ( x ) ) = x . See [link] , [link] , and [link] .
  • Only some of the toolkit functions have an inverse. See [link] .
  • For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
  • A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
  • For a tabular function, exchange the input and output rows to obtain the inverse. See [link] .
  • The inverse of a function can be determined at specific points on its graph. See [link] .
  • To find the inverse of a formula, solve the equation y = f ( x ) for x as a function of y . Then exchange the labels x and y . See [link] , [link] , and [link] .
  • The graph of an inverse function is the reflection of the graph of the original function across the line y = x . See [link] .

Section exercises

Verbal

Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that y -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no y -values repeat and the function is one-to-one.

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Why do we restrict the domain of the function f ( x ) = x 2 to find the function’s inverse?

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Can a function be its own inverse? Explain.

Yes. For example, f ( x ) = 1 x is its own inverse.

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Are one-to-one functions either always increasing or always decreasing? Why or why not?

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How do you find the inverse of a function algebraically?

Given a function y = f ( x ) , solve for x in terms of y . Interchange the x and y . Solve the new equation for y . The expression for y is the inverse, y = f 1 ( x ) .

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Algebraic

Show that the function f ( x ) = a x is its own inverse for all real numbers a .

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For the following exercises, find f 1 ( x ) for each function.

f ( x ) = x + 3

f 1 ( x ) = x 3

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f ( x ) = 2 x

f 1 ( x ) = 2 x

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f ( x ) = x x + 2

f 1 ( x ) = 2 x x 1

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f ( x ) = 2 x + 3 5 x + 4

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For the following exercises, find a domain on which each function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain.

f ( x ) = ( x + 7 ) 2

domain of f ( x ) : [ 7 , ) ; f 1 ( x ) = x 7

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f ( x ) = x 2 5

domain of f ( x ) : [ 0 , ) ; f 1 ( x ) = x + 5

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Given f ( x ) = x 2 + x and g ( x ) = 2 x 1 x :

  1. Find f ( g ( x ) ) and g ( f ( x ) ) .
  2. What does the answer tell us about the relationship between f ( x ) and g ( x ) ?

a.   f ( g ( x ) ) = x and g ( f ( x ) ) = x . b. This tells us that f and g are inverse functions

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Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
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Practice Key Terms 1

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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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