Inverse Of Quadratic Functions

Inverse Function of a Quadratic in Standard Form
Inverse Function of a Quadratic in Standard Form

Inverse Of Quadratic Functions

Find the inverse of quadratic functions with restricted domain; examples are presented along with with detailed solutions

Examples with Detailed Solutions

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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