A Gentle Introduction to Function Notation

A Gentle Introduction to Function Notation

Understanding f : A → B — the language of modern mathematics.

One of the most powerful ideas in mathematics is the concept of a function. We usually meet it in the form f(x) = 2x + 1, but the structure behind this idea is far richer. The notation 

f : A → B

captures the entire architecture of a function in a single line. In this article, we unpack this notation and explain exactly what it means, why it matters, and how it connects to the familiar expression f(x) = y.

1. What does f : A → B mean?

When we write

f : A → B

we are saying:

  • f is a function,
  • A is the domain — the set of inputs the function accepts,
  • B is the codomain — the set in which all outputs must lie.

In words:

A function assigns to every element of the domain A exactly one output in the codomain B.

Two rules always hold for genuine functions:

  • Every input must have an output.
  • No input may have more than one output.

Different inputs may share the same output (allowed), but one input cannot map to two different values (not allowed).

2. The domain and the codomain

Domain (A)

The domain tells us exactly which values we are allowed to feed into the function. If we write f : A → B, we are making a precise promise:

Only elements of A will be used as inputs for f.

Examples:

  • A = ℝ: the function accepts all real numbers.
  • A = {1, 2, 3}: the function is defined only for 1, 2 and 3.

Codomain (B)

The codomain is the set that contains all the possible outputs. It is not necessarily the set of outputs the function actually produces.

  • Codomain — where outputs are allowed to live.
  • Image (or range) — the outputs the function actually produces.

Example:

f(x) = x²,     f : ℝ → ℝ

Although the codomain is ℝ, the image consists only of non-negative real numbers.

3. Mapping notation: f : x ↦ y

We often describe a function by telling how it transforms a typical element. This is where the mapping arrow comes in: 

f : x ↦ y

This means:

“The function f sends the element x to the element y.”

This is simply another way of writing f(x) = y, but it highlights the fact that the function is a rule transforming one element into another.

Example:

f : x ↦ x²

This reads as: “take any input x and send it to x²”.

4. Combining domain, codomain and rule

A complete function definition often includes both the structural information and the mapping rule: 

f : A → B,   x ↦ y.

This tells us:

  • the function accepts elements from A,
  • its outputs belong to B,
  • and each input x is mapped to the output y.

Example:

f : ℝ → ℝ,   x ↦ x².

This defines a function on all real numbers that outputs the square of its input.

5. Images and preimages

If f(x) = y, we say:

  • y is the image of x under f,
  • x is a preimage of y.

This vocabulary becomes important when studying injective functions, surjective functions, bijections, and inverse functions.

6. Why this notation matters

The notation f : A → B is precise, compact and foundational. It tells you:

  • which inputs are valid,
  • what type of outputs the function produces,
  • whether two functions can be composed,
  • whether an inverse function might exist,
  • and how the function fits into a larger theory.

In calculus, analysis, linear algebra, topology and computer science, this notation is indispensable. Every derivative, projection, matrix transformation, and geometric mapping is simply a function written in this framework.

7. Examples

Example 1 — A linear function

f : ℝ → ℝ,   f(x) = 2x + 1.

This function accepts any real number and returns a real number.

Example 2 — A function on a finite set

g : {1, 2, 3} → {a, b, c, d}.

The function is defined only on the three inputs 1, 2 and 3.

Example 3 — A domain must sometimes be restricted

h(x) = 1/x.

The notation h : ℝ → ℝ is invalid because 1/0 is undefined. The correct definition is:

h : ℝ \ {0} → ℝ,   h(x) = 1/x.

Now the function is properly defined for every allowed input.

8. The complete picture

A function definition has three essential components:

  • The domain, where inputs come from.
  • The codomain, containing all possible outputs.
  • The mapping rule, telling us how each input is transformed.

The notation f : A → B captures the structure. The notation x ↦ y captures the action. The expression f(x) = y captures the result.

Together, they form the core language of functions — the foundation of nearly every idea in higher mathematics.

Notes prepared for personal study.

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