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 $f(x)=x^{2}$. 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.
• Shooting range
• Range - see distance and rangefinder in target and sniper shooting