Find the inverse of quadratic functions with restricted domain; examples are presented along with with detailed solutions

## Examples with Detailed Solutions

### 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 - √[ (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