# Difference between revisions of "Injection"

An injection, or "one-to-one function," is a function that takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the range is the image of exactly one value in the domain.

Alternative definition: A function $f:A\to B$ is an injection if for all $x,y\in A$, if $f(x)=f(y)$ then $x=y$.

## Examples

Linear functions are injections: $f:\mathbb R \to mathbb R$, $f(x)= ax+b$, $a\neq 0$. The domain choosing is also important. For example, while $f:\mathbb R \to \mathbb R$, $f(x)=x^2$ is not an injection ($f(-1)=f(1)=1$), the function $g:[0,\infty)\to\mathbb R$, $g(x)=x^2$, is an injection.