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 â€”
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) 

So, the inverse of f(x) = 12x  7 is
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.
