What is a Function?

A function is a rule that assigns exactly one output to each input.

Formal Definition

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

• Domain: The set of all possible inputs ($A$)
• Range: The set of all possible outputs ($B$)
• Codomain: The set $B$ (range ⊆ codomain)

Function Notation

$f(x) = x^2 + 3$

• $f$ is the name of the function
• $x$ is the input (also called the argument)
• $f(x)$ is the output (read as "f of x")

Evaluating Functions

To find $f(2)$ when $f(x) = x^2 + 3$:

$f(2) = (2)^2 + 3 = 4 + 3 = 7$

The Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph more than once.

✓ $y = x^2$ is a function (parabola passes test) ✗ $x^2 + y^2 = 1$ is not a function (circle fails test)

Domain Restrictions

Functions can have restricted domains:

Division by Zero

$f(x) = \frac{1}{x-2}$

Domain: All real numbers except $x = 2$ $\{x \in \mathbb{R} : x \neq 2\}$

Square Roots

$g(x) = \sqrt{x-3}$

Domain: $x \geq 3$ $\{x \in \mathbb{R} : x \geq 3\}$

Logarithms

$h(x) = \ln(x)$

Domain: $x > 0$ $\{x \in \mathbb{R} : x > 0\}$