Inverse Of Quadratic Functions

Examples, with detailed solutions, on how to find the inverse of quadratic functions with restricted domain are presented.

Example 1: Find the inverse of the quadratic function in vertex form given by

f(x) = 2(x - 2) 2 + 3 , for x <= 2

Solution to example 1:

  • Note that the above function is a quadratic function with restricted domain. Its graph below
    graph of quadratic function with restricted domain, example 1
    shows that it is a one to one function.Write the function as an equation.


    y = 2(x - 2) 2 + 3

  • Solve the above for x to obtain 2 solutions

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

    x - 2 = + or - sqrt[ (y - 3)/2 ]

    x = 2 + sqrt[ (y - 3)/2 ]

    and

    x = 2 - sqrt[ (y - 3)/2 ]

  • Since x given by x = 2 - sqrt[ (y - 3)/2 ] is always less than or equal to 2, we take the solution.

    x = 2 - sqrt[ (y - 3)/2 ]

  • Change x into y and y into x to obtain the inverse function.

    y = 2 - sqrt[ (x - 3)/2 ]

    f -1(x) = 2 - sqrt[ (x - 3)/2 ]

Example 2: Find the inverse of the quadratic function given by

f(x) = -2 x 2 + 4 x + 2 , for x >= 1

Solution to example 2:

  • We first need to show that this function is a one to one. Write f in vertex form by completing the square.

    f(x) = -2 (x 2 - 2 x) + 2 , for x >= 1

    f(x) = -2 (x 2 - 2 x + 1 - 1) + 2 , for x >= 1

    f(x) = -2 (x - 1) 2 + 4 , for x >= 1

    graph of quadratic function with restricted domain, example 2


  • The graph above is that of f and according to the horizontal line test f is a one to one function and therefore has an inverse.

  • Find the inverse of f, write f as an equation and solve for x. y = -2 (x - 1) 2 + 4

    x - 1 = + or - sqrt[ (y - 4)/- 2 ]

    x = 1 + sqrt[ (y - 4)/- 2 ]

    and

    x = 1 - sqrt[ (y - 4)/- 2 ]

  • Since x given by x = 1 + sqrt[ (y - 4)/- 2 ] is always greater than or equal to 1, we take the solution.

    x = 1 + sqrt[ (y - 4)/- 2 ]

  • Change x into y and y into x to obtain the inverse function.

    y = 1 + sqrt[ (x - 4)/- 2 ]

    f -1(x) = 1 + sqrt[ (x - 4)/- 2 ]




Exercises: Find the inverse of the quadratic functions given below

1. f(x) = (x - 3) 2 + 3 , if x >= 3

2. g(x) = -x 2 + 4 x - 4 , if x <= 2


Answers to above exercises:

1. f -1(x) = 3 + sqrt[ (x - 3) ]

2. g -1(x) = 2 - sqrt[ (-x) ]

More links and references related to the inverse functions.



Find the Inverse Functions - Calculator

Find inverse of exponential functions

Applications and Use of the Inverse Functions

Find the Inverse Function - Questions

Find the Inverse Function (1) - Tutorial.

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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