inverse function
An inverse function is the undo button. If f takes you from input to output, the inverse f-inverse takes that output and carries you back to the original input. Put on your shoes, then take them off — the second action reverses the first and returns you to where you started.
Written with notation, the inverse of f is denoted f^(-1), and it satisfies f^(-1)(f(x)) = x and f(f^(-1)(x)) = x. To find it from a rule, swap the roles of input and output: write y = f(x), interchange x and y, then solve for y. The result is the inverse rule.
Not every function has an inverse. Only one-to-one functions do, because the undo step needs a unique input to return to. The graph of an inverse is the mirror image of the original across the line y = x. And beware: f^(-1) means inverse, never the reciprocal 1/f.
For f(x) = 2x + 1: swap to x = 2y + 1, solve y = (x - 1)/2, so f^(-1)(x) = (x - 1)/2.
Swap, then solve — the undo rule.
The notation f^(-1) is a famous trap: it is the inverse function, not the power -1. The reciprocal 1/f(x) is a different thing entirely.