Undoing a Function

A function sends inputs to outputs. When each output comes from only one input, the function has an inverse that runs the other way: it sends each output back to the input it came from. For $f\left(x\right) = 3x$, f(4) is 12, so the inverse sends 12 back to 4. The inverse of f is written ${f}^{-1}$, and here ${f}^{-1}\left(x\right) = \frac{x}{3}$.

The -1 is not a power. ${f}^{-1}\left(x\right)$ means the inverse of f, never $\frac{1}{f\left(x\right)}$. For the rule above, ${f}^{-1}\left(12\right)$ is 4, while 1 divided by f(12) is 1/36.

Because the inverse swaps inputs and outputs, its table is f's table with the columns traded. And a function and its inverse undo each other: ${f}^{-1}\left(f\left(4\right)\right) = {f}^{-1}\left(12\right) = 4$. Whatever f does, its inverse takes back, like decoding a message that was encoded.