Inverse Functions
Understanding how to reverse a function — recovering the input from the output — and when this is possible.
1. The Idea of Reversing a Function
A function takes an input, applies a rule, and produces an output. An inverse function does the opposite: it starts with the output and recovers the original input.
Analogy: Think of a function as a machine. You feed in a number $x$, the machine applies a rule, and out comes $f(x)$. The inverse function $f^{-1}$ is a machine that takes $f(x)$ and spits the original $x$ back out.
For example, if $f(x) = 2x + 3$, then feeding in $x = 4$ gives $f(4) = 11$. The inverse function should take $11$ and return $4$. Indeed, $f^{-1}(x) = \dfrac{x - 3}{2}$, and $f^{-1}(11) = \dfrac{11 - 3}{2} = 4$. ✓
When is reversal possible? Only when every output came from exactly one input. If two different inputs produce the same output, the "reverse machine" wouldn't know which input to return — so no inverse exists (unless we restrict the domain).