Inverse Function
Functions & Advanced AlgebraAn 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 $7$, then the inverse takes $7$ and gives back $3$. They are reverse operations.
Example
$f(x) = x + 5$ adds $5$. Its inverse $f^{-1}(x) = x - 5$ subtracts $5$. If $f(3) = 8$, then $f^{-1}(8) = 3$. They undo each other.
Key Insight
Inverses swap inputs and outputs. On a graph, the inverse is the mirror image of the original across the line $y = x$.
Definition
The inverse function $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. 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 $y$.
Example
$f(x) = 2x - 6$. Step 1: $y = 2x - 6$. Step 2: swap: $x = 2y - 6$. Step 3: solve for $y$: $y = (x + 6)/2$. So $f^{-1}(x) = (x + 6)/2$. Check: $f(f^{-1}(4)) = f(5) = 4$. Correct.
Key Insight
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 $b^{\log_b(x)} = x$.
Definition
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 $f^{-1} \circ f = \text{id}_A$. In group theory, every element $g$ has an inverse $g^{-1}$ with $g \cdot g^{-1} = e$. The inverse function theorem guarantees a local inverse for differentiable functions with nonzero Jacobian determinant.
Example
The inverse function theorem: 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 $C^1$ inverse. This is used to prove the implicit function theorem.
Key Insight
Left vs. right inverses are distinct for non-bijective functions. In functional analysis, a bounded linear operator may have a left inverse (injective) or right inverse (surjective) but not both unless it is bijective.