Finding the Equation of an Inverse Function

The following procedure is helpful when trying to find the equation of the inverse of a function.

Procedure — To Find the Equation of the Inverse of a Function

If a function f(x) has an inverse, the following may be used to find the inverse function f ^-1(x).

Step 1 Replace f(x) with y.

Step 2 Switch the variables y and x.

Step 3 Solve for y.

Step 4 Replace y with f ^-1(x).

Note:

Why do we switch x and y when finding an inverse function?

 This is because if the function f(x) is satisfied by an ordered pair (a, b) then the inverse function f ^-1(x) is satisfied by the ordered pair (b, a).

Example

Given f(x) = 12x - 7, find f ^-1(x).

Solution

Step 1 Step 2 Step 3	Replace f(x) with y. Switch the variables y and x. Solve for y. Add 7 to both sides.	y x   x + 7	= 12x - 7 = 12y - 7   = 12y
	Divide both sides by 12.		= y
	We usually write y on the left.	y
Step 4	Replace y with f -1(x).	f-1(x)

Note:

Notice that in f(x) = 12x - 7, the input is multiplied by 12 and then 7 is subtracted from that product.

The reverse is true for the inverse. That is, in , 7 is added to the input and then that sum is divided by 12.