Injective Function¶

Definition¶

A function $f\colon X\to Y$ is injective if different elements map to different elements, i.e., for any $x,y\in X\text{,}$

Interpretation¶

An injective function never merges inputs: different inputs always give different outputs. Equivalently, each output comes from at most one input, so the value $f(x)$ is enough to recover $x$.

In the left diagram the three inputs go to three different points of $Y$, so the function is injective; one point of $Y$ is not reached at all, which injectivity allows. In the right diagram the inputs $x_2\neq x_3$ share an output, $f(x_2)=f(x_3)$$f(x_3)$, and this single collision is enough to make the function not injective.

In proofs the definition is mostly used in its contrapositive form: $f$ is injective if and only if, for all $x,y\in X$$X$,

So to prove injectivity, assume two outputs are equal and show that the inputs are equal; to disprove it, one pair suffices, as $(-1)^2=1^2$ shows for $x^2$ on $\mathbb{R}$. For a function on a subset of $\mathbb{R}$, injectivity means that every horizontal line meets the graph at most once.

Injectivity depends on the domain: $x^2$ is not injective on $\mathbb{R}$, but it is injective on $[0,+\infty)$, and there its inverse is $\sqrt{x}$. Every strictly monotone function is injective, which gives a quick sufficient test. Together with surjectivity, injectivity makes $f$ a bijection with an inverse.

A common slip is to check the wrong implication. The statement $x=y\implies f(x)=f(y)$ holds for every function and says nothing about injectivity.

Key Results¶

• Injectivity depends on both formula and domain.

Examples¶

• $f(x)=x^3$ is injective on $\mathbb{R}\text{.}$
• $f(x)=x^2$ is not injective on $\mathbb{R}\text{.}$

Prerequisites¶

• Domain

Related Concepts¶

• Surjective Function
• Monotone Function

Sources¶

• Math I: Calculus · Lecture 4

Aliases: one-to-one function, injection, injective Symbols: element of, infinity, real numbers, square root

Last updated: 20 October 2026 03:02