Inverse Of Quadratic Functions
Find the inverse of quadratic functions with restricted domain; examples are presented along with with detailed solutions
Examples with Detailed SolutionsExample 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  √[ (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

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.
Find the Inverse Functions  Calculator
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Definition of the Inverse Function  Interactive Tutorial
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