Find Inverse Function (1) - Tutorial

Examples on how to find inverse functions analytically are presented. Detailed solutions and matched exercises with answers at the end of this page are also included.

Example 1: Find the inverse function of f given by

f(x) = 2x + 3
Solution to example 1:
  • write the function as an equation.
      y = 2x + 3

  • solve for x.
      x = (y - 3)/2

  • now write f-1(y) as follows .
      f -1(y) = (y - 3)/2
      or
      or f -1(x) = (x - 3)/2

  • Check
    1. f(f -1(x))=2(f -1(x)) + 3

        =2((x-3)/2)+3

        =(x-3)+3

        =x

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

        =((2x+3)-3)/2

        =2x/2

        =x

conclusion: The inverse of function f given above is f -1(x) = (x - 3)/2

Matched Exercise 1: Find the inverse function of f given by

f(x) = -x - 4


Example 2: Find the inverse function of f given by

f(x) = (x - 3)2, if x >= 3
Solution to example 2:
  • write the function as an equation.
      y = (x - 3)2


  • solve for x, two solutions .
      x = 3 + sqrt(y)

      x = 3 - sqrt(y)

  • the first solution is selected since x >= 3, write f-1(y) as follows.

      f -1(y) = 3 + sqrt(y)

      or

      f -1(x) = 3 + sqrt(x)

  • Check
      f(f -1(x))=((3+sqrt(x))-3)2

      =(sqrt(x))2

      =x

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

      =3+|x-3| (since x >= 3, x-3 >= 0, |x-3| = x-3)

      =3+(x-3)

      =x

conclusion:
The inverse of function f given above is f -1(x) = 3 + sqrt(x)

Matched Exercise 2: Find the inverse function of f given by
f(x) = (x + 1)2, if x >= -1


Example 3: Find the inverse function of f given by

f(x) = (x + 1)/(x - 2)
Solution to example 3:
  • Write the function as an equation.
      y = (x + 1) / (x - 2)


  • Multiply both sides of the above equation by x - 2 and simplify.
      y (x - 2) = x + 1

  • Multiply and group.

      y x - 2y = x + 1

      y x - x = 2y + 1

  • Factor x on the left side and solve
      x(y - 1) = 1 + 2y

      x = (1 + 2y) / (y - 1)

  • Change x to y and y to x
    y = (1 + 2x) / (x - 1)
  • The inverse of function f given above is
    f -1(x) = (1 + 2x) / (x - 1)


Matched Exercise 3: Find the inverse function of f given by

f(x) = (x + 1)/(x - 1


Answers to the Matched Exercises

Answer to Matched Exercise 1

f -1(x) = -x - 4

Answer to Matched Exercise 2

f -1(x) = -1 + sqrt(x)

Answer to Matched Exercise 3

f -1(x) = (x + 1)/(x - 1)

More links and references related to the inverse functions.

Find the Inverse Functions - Calculator

Applications and Use of the Inverse Functions

Find the Inverse Function - Questions

Inverse of Quadratic Functions.

Definition of the Inverse Function - Interactive Tutorial

Find Inverse Of Cube Root Functions.

Find Inverse Of Square Root Functions.

Find Inverse Of Logarithmic Functions.

Find Inverse Of Exponential Functions.


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Updated: 2 April 2013

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