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 1Find the inverse of the quadratic function in vertex form given by 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 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 ]
## ExercisesFind the inverse of the quadratic functions given below
Answers to Above Exercises
More links and references related to the inverse functions. |