This section explains how to determine if the inverse of a function is a function. A graphing calculator is needed.

Example: Is the inverse of f (x) = x2 + 4 a function?

f(x) = x2 + 4
y = x2 + 4
x = y2 + 4
y2 = x – 4
y = ± √x – 4

Rewrite the equation using y.
Switch x and y.
Isolate y2.
Take the square root of both sides.
Use your graphing calculator to graph both functions
(y1 = √x – 4 and y2 = –√x – 4).
Test the function using the vertical and horizontal line test.

Interactive popup. Assistance may be required.

Check Your Answer

The inverse is not a function since it does not pass the vertical line test.Close Pop Up

Example: Is the inverse of f (x) = √x + 8 a function?

f(x) = √x + 8
y = √x + 8
x = √y + 8
x2 = (√y + 8)2
x2 = y + 8
y = x2 – 8

Rewrite the equation using y.
Switch x and y.
Square both sides.
Isolate y.
Use your graphing calculator to graph the function y1 = x2 – 8.

Interactive popup. Assistance may be required.

Check Your Answer

The inverse is a function since it passes the vertical line test.Close Pop Up

 

More Examples

Example Is the inverse of f(x) = √x + 5 a function?

f(x) = √x + 5
y = √x + 5
x = √y + 5
x – 5 = √y
(x – 5)2 = (√y)2
x2 – 10x + 25 = y
y = x2 – 10x + 25

Rewrite the equation using y.
Switch x and y.
Isolate √y.
Squareboth sides.

Isolate y.

Use your graphing calculator to graph the function
y1 = x2 – 10x + 25.

Interactive popup. Assistance may be required.

Check Your Answer

The inverse is a function since it passes the vertical line test.Close Pop Up

 

Example: Is the inverse of f (x) = 9x2 + 36 a function?

f(x) = 9x2 + 36
y = 9x2 + 36
x = 9y2 + 36
9y2 = x – 36
9y2 9 = x – 36 9
y2 = x – 36 9
y = + or – square root of (x over 9 less 4)

Rewrite the equation using y.
Switch x and y.
Isolate y2.






Take the square root of both sides.

Use your graphing calculator to graph both functions

Interactive popup. Assistance may be required.

Check Your Answer

The inverse is not a function since it does not pass the vertical line test.Close Pop Up