Bijective function

A function $f:X\to Y$ is bijective if it is injective and surjective ; equivalently,

where $\exists!$ means “there exists exactly one.”

Bijectivity is the precise condition for the existence of an inverse function $f^{-1}:Y\to X$ satisfying $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$.

Examples:

• $f:\mathbb{R}\to\mathbb{R}$, $f(x)=x+1$ is bijective.
• $\exp:\mathbb{R}\to(0,\infty)$ is bijective.
• $x\mapsto x^2$ is not bijective from $\mathbb{R}$ to $\mathbb{R}$ (not injective and not surjective), but it is bijective from $[0,\infty)$ to $[0,\infty)$.