Injective function

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 $$x_1 \neq x_2$$ implies that $$f(x_1) \neq f(x_2)$$.

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).