143 Inverse Function
An inverse function is a mapping that reverses another mapping that is used to recover inputs from outputs when a function is bijective.
Definition (Inverse Function) A function from the codomain of a function to its domain,
\[ f^{-1} : Y \rightarrow X, \]
where the following condition applies:
- \(f^{-1}(y) = x \iff f(x) = y\).
where
- \(f : X \rightarrow Y\) is a one-to-one and onto function.
- \(X\) is the domain of \(f\).
- \(Y\) is the codomain of \(f\).
- \(x \in X, y \in Y\).
143.1 Elementary Example
143.1.1 Simple
An inverse function reverses a bijection: \(f^{-1}(y) = x\) exactly when \(f(x) = y\).
\[ f : X \rightarrow Y \]
\[ X = \{ 1,\ 2,\ 3 \},\quad Y = \{ a,\ b,\ c \} \]
\[ f(1)=a,\ f(2)=b,\ f(3)=c \]
\[ f^{-1}(a)=1,\ f^{-1}(b)=2,\ f^{-1}(c)=3 \]
where
- \(f\) is bijective.
- \(f^{-1} : Y \rightarrow X\) is the inverse function.