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(x) = 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(x) = x/3.

The -1 is not a power. f^-1(x) means the inverse of f, never 1/(f(x)). For the rule above, f^-1(12) 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(f(4)) = f^-1(12) = 4. Whatever f does, its inverse takes back, like decoding a message that was encoded.