Function Basics

Relation vs. Function

VisualEvery input on the x-axis shoots one arrow upward. If two arrows ever land on the same x but different y, it's just a relation — not a function.

AlgebraA function is a relation where each input $x$ maps to exactly one output $f(x)$. The vertical line test catches violations visually.

• • Passes vertical line test → function
• • f(x) reads 'f of x', not 'f times x'

Domain

VisualSweep a vertical line across the grid. The x-values where the graph exists form the domain.

Algebra$\text{dom}(f) = \{x \in \mathbb{R} : f(x) \text{ is defined}\}$

• • Watch for: division by zero, even roots of negatives, logs of non-positives

Range

VisualSweep a horizontal line. The y-values the graph reaches form the range.

Algebra$\text{ran}(f) = \{f(x) : x \in \text{dom}(f)\}$

Function Notation

VisualDrop the point $a$ onto the curve — the height $f(a)$ is the function's value at $a$.

AlgebraIf $f(x)=x^2-2$, then $f(3) = 3^2-2 = 7$.