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Inverse function

The function that undoes a bijective function.
Inverse function

Let f:X→Yf:X\to Y be . The inverse function of ff is the f−1:Y→Xf^{-1}:Y\to X defined by

f−1(y)=xwhere x∈X is the unique element with f(x)=y.f^{-1}(y)=x\quad\text{where $x\in X$ is the unique element with }f(x)=y.

It satisfies

f−1∘f=idXandf∘f−1=idY,f^{-1}\circ f = \mathrm{id}_X \quad\text{and}\quad f\circ f^{-1}=\mathrm{id}_Y,

where idX:X→X\mathrm{id}_X:X\to X is the identity map idX(x)=x\mathrm{id}_X(x)=x (see ).

Inverse functions are central in analysis: many theorems (e.g., inverse function theorems) give conditions under which a function has a (local) inverse with additional regularity.

Examples:

  • If f:R→Rf:\mathbb{R}\to\mathbb{R}, f(x)=x+1f(x)=x+1, then f−1(y)=y−1f^{-1}(y)=y-1.
  • exp⁡:R→(0,∞)\exp:\mathbb{R}\to(0,\infty) has inverse log⁡:(0,∞)→R\log:(0,\infty)\to\mathbb{R}.
  • The function x↦x2x\mapsto x^2 has no inverse as a map R→R\mathbb{R}\to\mathbb{R}, but it has an inverse on [0,∞)[0,\infty) given by ⋅:[0,∞)→[0,∞)\sqrt{\cdot}:[0,\infty)\to[0,\infty).