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In mathematics, the range (or image) of a function are the values it hits. It is not to be confused with the codomain of a function, which is a designated set to which all the values of the function belong.

A function is onto (or surjective) if every value in its codomain is hit by the function, or, equivalently, if its range is equal to its codomain. More formally, a function f: A \to B is onto if for every y \in B there exists x \in A such that f(x) = y.


Let f: \mathbb{R} \to \mathbb{R} be the function defined by the equation f(x) = x2. By definition, the codomain of f is \mathbb{R}. However, the range of f consists of all nonnegative real numbers. Indeed, let y be a nonnegative real number. Then f(\sqrt{y}) = y, and so y is one of the values hit by f.

Let g: \mathbb{R} \to \mathbb{R} be the function defined by the equation g(x) = x + 1. Then, for every real number y, we can see that g(y − 1) = (y − 1) + 1 = y, so every real number is hit by g. This means that the codomain and range of g are equal, namely \mathbb{R}. Therefore, g is onto.

Non-mathematical uses

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