# Injective function

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In mathematics, an injective function or one-to-one function or injection is a function which has different output values on different input values: f is injective if ${\displaystyle x_{1}\neq x_{2}}$ implies that ${\displaystyle f(x_{1})\neq f(x_{2})}$.
An injective function f has a well-defined partial inverse ${\displaystyle f^{-1}}$. If y is an element of the image set of f, then there is at least one input x such that ${\displaystyle f(x)=y}$. If f is injective then this x is unique and we can define ${\displaystyle f^{-1}(y)}$ to be this unique value. We have ${\displaystyle f^{-1}(f(x))=x}$ for all x in the domain.
A strictly monotonic function is injective, since in this case ${\displaystyle x_{1} implies that ${\displaystyle f(x_{1}).