Injective function

A function that never maps two distinct inputs to the same output.

Injective function

A function $f:X\to Y$ is injective (or one-to-one) if

\forall x_1,x_2\in X,\ \bigl(f(x_1)=f(x_2)\Rightarrow x_1=x_2\bigr).

Injectivity means that outputs uniquely determine inputs (within the domain ). This is the exact condition needed for a (two-sided) inverse function to exist after restricting the codomain to the image .

Examples:

$f:\mathbb{R}\to\mathbb{R}$ , $f(x)=x^3$ is injective.
$f:\mathbb{R}\to\mathbb{R}$ , $f(x)=x^2$ is not injective since $f(1)=f(-1)=1$ .
The restriction $x\mapsto x^2$ on $[0,\infty)$ is injective.