Swap and Solve

To find an inverse rule, trade the roles of input and output. Take $f\left(x\right) = 4x+8$ and write it with y for the output: $y = 4x+8$. Then swap x and y: $x = 4y+8$. This equation says the same thing with input and output traded.

Now solve for y, since y is the new output. Subtract 8: $x-8 = 4y$. Divide by 4 and split the fraction: $y = \frac{x}{4}-2$. So ${f}^{-1}\left(x\right) = \frac{x}{4}-2$.

Check it with one number. f(3) is 20, and the inverse sends 20 to 5 minus 2, which is 3. The swap is the whole idea of an inverse; the solving after it is ordinary algebra.