# Range

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.

## Examples

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

A range can also refer to a type of oven.