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

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 ]

Find the inverse of the quadratic function given by
f(x) = -2 x² + 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 (x² - 2 x) + 2 , for x >= 1
f(x) = -2 (x² - 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 ]

Find the inverse of the quadratic functions given below
1. f(x) = (x - 3)² + 3 , if x >= 3
2. g(x) = -x² + 4 x - 4 , if x <= 2

Answers to Above Exercises
1. f^-1(x) = 3 + √[ (x - 3) ]
2. g^-1(x) = 2 - √[ (-x) ]