How to Invert a Function to Find Its Inverse
If you’re given a function and must find its inverse, first remind yourself that domain and range swap places in the functions. Literally, you exchange f(x) and x in the original equation. When you make that change, you call the new f(x) by its true name — f–1(x) — and solve for this function.
For example, follow the steps to find the inverse of this function:
Switch f(x) and x.
When you switch f(x) and x, you get
(Note: To make the notation less clumsy, you can rewrite f(x) as y and then switch x and y.)
Change the new f(x) to its proper name — f–1(x).
The equation then becomes
Solve for the inverse.
This step has three parts:
Multiply both sides by 3 to get 3x = 2f–1(x) –1.
Add 1 to both sides to get 3x + 1 = 2f–1(x).
Lastly, divide both sides by 2 to get your inverse: