# Injective function

From Knowino

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 implies that .

An injective function *f* has a well-defined partial inverse *f*^{ − 1}. If *y* is an element of the image set of *f*, then there is at least one input *x* such that *f*(*x*) = *y*. If *f* is injective then this *x* is unique and we can define *f*^{ − 1}(*y*) to be this unique value. We have *f*^{ − 1}(*f*(*x*)) = *x* for all *x* in the domain.

A strictly monotonic function is injective, since in this case *x*_{1} < *x*_{2} implies that *f*(*x*_{1}) < *f*(*x*_{2}) (if *f* is increasing) or *f*(*x*_{1}) > *f*(*x*_{2}) (if *f* is decreasing).

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