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
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

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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