Functions

A function is one of the most fundamental concepts in mathematics. It describes a precise relationship between two sets of numbers.

What Is a Function?

📐Definition — Function

A function $f$ from set $A$ to set $B$ is a rule that assigns to each element $x \in A$ exactly one element $y \in B$ . We write $y = f(x)$ .

$x$ is called the input (or argument)
$y = f(x)$ is called the output (or value)

The key word is exactly one. For every valid input, the function produces one and only one output. The notation $f(2)$ means “evaluate the function $f$ at $x = 2$ .”

Domain and Range

📐Definition — Domain and Range

The domain of $f$ is the set of all valid inputs $x$ for which $f(x)$ is defined.
The codomain is the target set $B$ .
The range is the set of all actual outputs: $\{f(x) : x \in \text{domain}\}$ .

The range is always a subset of the codomain, but they are not always equal.

Common Examples

Function	Formula	Domain
Linear	$f(x) = 2x + 1$	all real numbers
Quadratic	$f(x) = x^2$	all real numbers
Square root	$f(x) = \sqrt{x}$	$x \geq 0$
Reciprocal	$f(x) = \dfrac{1}{x}$	$x \neq 0$

✎Example — Evaluating a Function

Let $f(x) = 2x^2 - 3x + 1$ . Find $f(3)$ .

Solution. Substitute $x = 3$ :

$f(3) = 2(3)^2 - 3(3) + 1 = 18 - 9 + 1 = 10$

The Vertical Line Test

A graph in the $xy$ -plane represents a function if and only if every vertical line intersects the graph at most once. If any vertical line crosses the graph in two or more points, the graph fails the test — it does not define a function, because one input would map to multiple outputs.

For instance, the graph of $y = x^2$ passes the test (parabola), but the graph of $x^2 + y^2 = 1$ (circle) does not.

Piecewise Functions

A piecewise function uses different formulas on different parts of its domain.

✎Example — Piecewise Function

f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ 2x + 1 & \text{if } x \geq 0 \end{cases}

Evaluate $f(-2)$ and $f(3)$ .

Solution.

Since $-2 < 0$ , use the first rule: $f(-2) = (-2)^2 = 4$ .
Since $3 \geq 0$ , use the second rule: $f(3) = 2(3) + 1 = 7$ .

Piecewise functions are especially useful for modeling real-world situations where different rules apply under different conditions — for example, tax brackets or shipping rates.