Range and Image¶

Definition¶

If the function is $f\colon X\to Y\text{,}$ the range or image of $f$ is the set

We call an output $y=f(x)$ the dependent variable.

Interpretation¶

The range, or image, of a function is the set of outputs that actually occur. The codomain only says where outputs are allowed to lie; the range records which of these values the function really produces.

For $f(x)=x^2$ with codomain $\mathbb{R}$, every output is a square, and every $y\ge0$ is reached, at $x=\sqrt{y}$$\sqrt{y}$: the range is the red half-line $[0,+\infty)$ on the $y$-axis, while negative numbers lie in the codomain but are never taken.

To decide whether $y$ lies in the range, solve $f(x)=y$ for $x$ in the domain: $y$ belongs to the range if and only if there is a solution. The range equals the codomain exactly when $f$ is surjective. Some texts use the word "range" for the codomain; "image" avoids the ambiguity, and Math III writes it as $f(X)$, the image of the whole domain.

Key Results¶

• The range is a subset of the codomain.
• Different functions may share the same range.
• Determining the range often requires solving equations or inequalities.

Examples¶

• The range of $f(x)=x^2$ is $[0,+\infty)\text{.}$
• The range of $f(x)=\sin x$ is $[-1,1]\text{.}$

Prerequisites¶

• Function

Related Concepts¶

• Codomain
• Domain
• Image and Preimage of a Set
• Bijection and Inverse Function
• Range of a Linear Map

Sources¶

• Math I: Calculus · Lecture 4

Aliases: image of a function Symbols: element of, infinity, real numbers, square root

Last updated: 20 October 2026 03:02