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 - √[ (y - 3)/2 ]
    x = 2 + √[ (y - 3)/2 ]
    and
    x = 2 - √[ (y - 3)/2 ]
  • Since x given by x = 2 - √[ (y - 3)/2 ] is always less than or equal to 2, we take the solution.
    x = 2 - √[ (y - 3)/2 ]
  • Change x into y and y into x to obtain the inverse function.
    y = 2 - √[ (x - 3)/2 ]
    f -1(x) = 2 - √[ (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 - √[ (y - 4)/- 2 ]
    x = 1 + √[ (y - 4)/- 2 ]
    and
    x = 1 - √[ (y - 4)/- 2 ]
  • Since x given by x = 1 + √[ (y - 4)/- 2 ] is always greater than or equal to 1, we take the solution.
    x = 1 + √[ (y - 4)/- 2 ]
  • Change x into y and y into x to obtain the inverse function.
    y = 1 + √[ (x - 4)/- 2 ]
    f -1(x) = 1 + √[ (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 + √[ (x - 3) ]
2. g -1(x) = 2 - √[ (-x) ]

More links and references related to the inverse functions.
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Definition of the Inverse Function - Interactive Tutorial
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