Bijective function
A bijective function is both injective and surjective. It pairs the domain and codomain perfectly: every input has exactly one output, every codomain element is hit, and no two inputs collide.
Formally, is bijective when for every there exists a unique such that .
Examples
The function
\qquad f(x)=2x+1$$ is bijective. Its inverse is $$f^{-1}(y)=\frac{y-1}{2}.$$ The function $x^2$ is not bijective as $\mathbb R\to\mathbb R$. It fails injectivity and misses negative outputs. But $$g:[0,\infty)\to[0,\infty), \qquad g(x)=x^2$$ is bijective, with inverse $g^{-1}(y)=\sqrt y$. ## Why it matters Bijectivity is exactly the condition for a two-sided [[Inverse functions|inverse function]]. It means the mapping loses no information and includes no unreachable target values. Bijective mappings are used to show that two sets have the same cardinality, to change coordinates without losing information, and to move between equivalent descriptions of a system. In physics, a reversible change of coordinates should be bijective on the relevant state space. Related: [[Function]], [[Mapping]], [[Codomain and range]], [[Domain]].