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Finding inverses of rational functions

As with finding inverses of quadratic functions, it is sometimes desirable to find the inverse of a rational function , particularly of rational functions that are the ratio of linear functions, such as in concentration applications.

Finding the inverse of a rational function

The function C = 20 + 0.4 n 100 + n represents the concentration C of an acid solution after n mL of 40% solution has been added to 100 mL of a 20% solution. First, find the inverse of the function; that is, find an expression for n in terms of C . Then use your result to determine how much of the 40% solution should be added so that the final mixture is a 35% solution.

We first want the inverse of the function in order to determine how many mL we need for a given concentration. We will solve for n in terms of C .

C = 20 + 0.4 n 100 + n C ( 100 + n ) = 20 + 0.4 n 100 C + C n = 20 + 0.4 n 100 C 20 = 0.4 n C n 100 C 20 = ( 0.4 n C ) n n = 100 C 20 0.4 C

Now evaluate this function at 35%, which is C = 0.35.

n = 100 ( 0.35 ) 20 0.4 0.35 = 15 0.05 = 300

We can conclude that 300 mL of the 40% solution should be added.

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Find the inverse of the function f ( x ) = x + 3 x 2 .

f 1 ( x ) = 2 x + 3 x 1

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Key concepts

  • The inverse of a quadratic function is a square root function.
  • If f 1 is the inverse of a function f , then f is the inverse of the function f 1 . See [link] .
  • While it is not possible to find an inverse of most polynomial functions, some basic polynomials are invertible. See [link] .
  • To find the inverse of certain functions, we must restrict the function to a domain on which it will be one-to-one. See [link] and [link] .
  • When finding the inverse of a radical function, we need a restriction on the domain of the answer. See [link] and [link] .
  • Inverse and radical and functions can be used to solve application problems. See [link] and [link] .

Section exercises

Verbal

Explain why we cannot find inverse functions for all polynomial functions.

It can be too difficult or impossible to solve for x in terms of y .

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Why must we restrict the domain of a quadratic function when finding its inverse?

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When finding the inverse of a radical function, what restriction will we need to make?

We will need a restriction on the domain of the answer.

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The inverse of a quadratic function will always take what form?

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Algebraic

For the following exercises, find the inverse of the function on the given domain.

f ( x ) = ( x 4 ) 2 ,   [ 4 , )

f 1 ( x ) = x + 4

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f ( x ) = ( x + 2 ) 2 ,   [ −2 , )

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f ( x ) = ( x + 1 ) 2 3 ,   [ −1 , )

f 1 ( x ) = x + 3 1

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f ( x ) = 3 x 2 + 5 , ( , 0 ]

f 1 ( x ) = x 5 3

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f ( x ) = 12 x 2 ,   [ 0 , )

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f ( x ) = 9 x 2 ,   [ 0 , )

f ( x ) = 9 x

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f ( x ) = 2 x 2 + 4 ,   [ 0 , )

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For the following exercises, find the inverse of the functions.

f ( x ) = x 3 + 5

f −1 ( x ) = x 5 3

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f ( x ) = 4 x 3

f 1 ( x ) = 4 x 3

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For the following exercises, find the inverse of the functions.

f ( x ) = 2 x + 1

f −1 ( x ) = x 2 1 2 , [ 0 , )

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f ( x ) = 9 + 4 x 4

f −1 ( x ) = ( x 9 ) 2 + 4 4 , [ 9 , )

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Questions & Answers

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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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