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.

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Injection

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