Definition:Real Function/Range

This page is about the range of a real function. For other uses, see range.

Definition

Let $S \subseteq \R$.

Let $f: S \to \R$ be a real function.

The range of $f$ is the set of values that the dependent variable can take.

That is, it is the image set of $f$.

Examples

Arbitrary Example

Let $S$ be the closed real interval $\closedint {-1} 1$.

Let $f: S \to \R$ be the function defined as:

$\forall x \in S: \map f x = 2 x$

Then the range of $f$ is $\closedint {-2} 2$.

Also known as

Modern treatments of the subject are usually more precise.

In particular, the word range may be taken to mean codomain.

Also see

• Results about ranges of real functions can be found here.

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term Function of One Independent Variable
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): range: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): range: 1.