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. If you wish to first review the definition and some properties of the inverse function Go Here.

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.

Applications and Use of the Inverse Functions

Find the Inverse Function - Questions

Inverse of Quadratic Functions.

Definition of the Inverse Function - Interactive Tutorial

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Updated: 27 November 2007 (A Dendane)