Inverse Functions

Finding the Inverse of a Function

Given the function f(x) we want to find the inverse function, f−1(x).

  • Replace f(x) with y.
  • Just rearrange to get x =?
  • Then swap the “x”s and “y”s.
  • Finally replace y with f−1(x).


Example Given f(x) = 3x−2 find f−1(x).

First y = 3x−2

Next, rearrange to get x. x = y/3 −2/3

Then swap the “x”s and “y”s. y = x/3+2/3

Finally replace y with f−1(x). f−1(x) = x/3+2/3


Example Given f(x) = (5x-2)/(2x+3) find f−1(x).

First y = (5x-2)/(2x+3)

Next, rearrange to get x. y(2x+3) = 5x-2

2yx +3y = 5x-2

2yx -5x = -2-3y

x(2y-5) = -2-3y

x = (-2-3y) / (2y-5)

Then swap the “x”s and “y”s. y = (-2-3x) / (2x-5)

Finally replace y with f−1(x). f−1(x) = (-2-3x) / (2x-5)