Image of a Function — Definition, Formula & Examples

The image of a function is the set of all output values the function actually produces. If you feed every element of the domain into the function, the collection of results you get is the image.

Given a function $f: A \to B$ , the image of $f$ is the set $\text{im}(f) = \{f(x) \mid x \in A\} \subseteq B$ . The image is a subset of the codomain consisting of exactly those elements in $B$ that are mapped to by at least one element in $A$ .

Key Formula

\text{im}(f) = \{ f(x) \mid x \in A \}

Where:

$f$ = The function from set A to set B
$A$ = The domain (set of all inputs)
$x$ = An element of the domain
$\text{im}(f)$ = The image — the set of all output values

How It Works

To find the image, determine which output values are reachable. For a formula like

f(x) = x^2

with domain

\mathbb{R}

, every input squares to a non-negative number, so the image is

[0, \infty)

. For a finite set, just apply the function to each input and collect the results. The image can be smaller than the codomain — when they are equal, the function is called surjective (onto).

Worked Example

Problem: Let f: {1, 2, 3, 4} → ℤ be defined by f(x) = 2x − 1. Find the image of f.

Step 1: Apply the function to each element of the domain.

f(1) = 1,\quad f(2) = 3,\quad f(3) = 5,\quad f(4) = 7

Step 2: Collect all the output values into a set.

\text{im}(f) = \{1, 3, 5, 7\}

Answer: The image of f is {1, 3, 5, 7}.

Why It Matters

Finding the image tells you the actual range of a function, which is essential when composing functions or determining if a function is surjective. In precalculus and calculus, identifying the image helps you sketch graphs, find inverse functions, and understand transformations.

Common Mistakes

Mistake: Confusing the image (range) with the codomain.

Correction: The codomain is the set a function is declared to map into; the image is the subset of the codomain that actually gets hit. For f(x) = x² with codomain ℝ, the codomain is all of ℝ but the image is only [0, ∞).