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
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Applications and Use of the Inverse Functions
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Find the Inverse Function (1)  Tutorial.
Definition of the Inverse Function  Interactive Tutorial
Find Inverse Of Cube Root Functions.
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Find Inverse Of Exponential Functions.
