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# 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 1Step 2 Step 3 Replace f(x) with y. Switch the variables y and x. Solve for y. Add 7 to both sides. yx   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.