Inverse Function
An inverse function reverses the effect of the original function, swapping inputs and outputs so that f(f^-1(x)) = x.
Formula
f(f^{-1}(x)) = f^{-1}(f(x)) = x
Definition
An inverse function undoes what the original function did: if $f$ takes $3$ and gives $7$, the inverse takes $7$ and gives back $3$, swapping inputs and outputs. The inverse function $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$, but a function has an inverse only if it is one-to-one (passes the horizontal line test); to find $f^{-1}$, replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$. Formally, a bijection $f: A \to B$ has a unique two-sided inverse $f^{-1}: B \to A$ satisfying $f \circ f^{-1} = \text{id}_B$ and $f^{-1} \circ f = \text{id}_A$; the inverse function theorem guarantees a local inverse for differentiable functions with nonzero Jacobian determinant.
Example
$f(x) = x + 5$ adds $5$, so its inverse $f^{-1}(x) = x - 5$ subtracts $5$: if $f(3) = 8$, then $f^{-1}(8) = 3$. For $f(x) = 2x - 6$: write $y = 2x - 6$, swap to $x = 2y - 6$, then solve to get $f^{-1}(x) = (x + 6)/2$; checking, $f(f^{-1}(4)) = f(5) = 4$. The inverse function theorem states that if $f: \mathbb{R}^n \to \mathbb{R}^n$ is $C^1$ and $\det(Df(p)) \neq 0$, then $f$ is locally invertible near $p$ with a $C^1$ inverse, a result used to prove the implicit function theorem.
Key Insight
On a graph, the inverse is the mirror image of the original across the line $y = x$, and the domain of $f^{-1}$ is the range of $f$ and vice versa; exponential and logarithmic functions are inverses of each other, $\log_b(b^x) = x$ and $b^{\log_b(x)} = x$. Left and right inverses are distinct for non-bijective functions: in functional analysis, a bounded linear operator may have a left inverse (if injective) or a right inverse (if surjective) but not both unless it is bijective.