Fractions in Inverses

When the rule divides, the inverse multiplies. Take $f\left(x\right) = \frac{x+6}{2}$. Write $y = \frac{x+6}{2}$ and swap: $x = \frac{y+6}{2}$.

Clear the fraction first by multiplying both sides by 2: $2x = y+6$. Then subtract 6: $y = 2x-6$. So ${f}^{-1}\left(x\right) = 2x-6$. Check: f(4) is 5, and the inverse sends 5 to 10 minus 6, which is 4.

Notice the order. f adds 6, then halves. The inverse doubles, then takes 6 away: each step undone, in reverse order, like taking off shoes before socks.