Function Properties: One-To-One

Definition. [One-To-One Functions] A function is called one-to-one or injective if different elements in the domain always map to different elements in the range. Symbolically, a function $f$ is injective if $f(x_1) = f(x_2)$ implies that $x_1 = x_2$.

Equivalently, in a one-to-one function, if $x_1 \ne x_2$, then $f(x_1)$ is never equal to $f(x_2)$.

For example, the function $f(x) = 2x + 3$ from $\mathbb{R} \to \mathbb{R}$ is injective.

Not every function is injective, however. The function $f(x) = x^2$ from $\mathbb{R} \to \mathbb{R}$ is not injective, because $f(2) = f(-2)$ (and, clearly, $2 \neq -2$.)

Injectivity ensures that no two domain elements share the same output (= range) value.