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.
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.
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.
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
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