# Inverse Of Quadratic Functions

Inverse Function of a Quadratic in Standard Form
Inverse Function of a Quadratic in Standard Form

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 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.

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