# 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*)*f*^{–1}(*x*).The equation then becomes

Solve for the inverse.

This step has three parts:

Multiply both sides by 3 to get 3

*x*= 2*f*^{–1}(*x*) –1.Add 1 to both sides to get 3

*x*+ 1 = 2*f*^{–1}(*x*).Lastly, divide both sides by 2 to get your inverse: