Finding the Equation of an Inverse Function
The following procedure is helpful when trying to find the equation of the
inverse of a function.
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).
Why do we switch x and y when finding an
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).
Given f(x) = 12x - 7, find f -1(x).
So, the inverse of f(x) = 12x - 7 is
|Replace f(x) with y.
Switch the variables y and x.
Solve for y.
Add 7 to both sides.
x + 7
|= 12x - 7
= 12y - 7
||Divide both sides by 12.
||We usually write y on the left.
||Replace y with f -1(x).
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
added to the input and then that
sum is divided by 12.