Graphs of Functions

Definition

The graph of a function $f$ is the set of all ordered pairs $(x, f(x))$ in the coordinate plane where $x$ is in the domain of $f$. Formally: $\text{Graph}(f) = \{ (x, y) \mid x \in D, y = f(x) \}$

• How to read: “The graph of f is the set of all ordered pairs x y such that x is in the domain and y equals f of x.”
• Meaning: Every point on the graph is a valid input-output pair; plotting all of them draws the function’s curve.

Why It Matters

Graphs transform abstract algebraic rules into intuitive spatial patterns; they allow us to ‘see’ the behavior of a system—identifying trends, symmetries, and discontinuities at a glance—which is essential for debugging models and communicating complex data.

Core Concepts

• Visual Mapping: The graph provides a geometric representation of the algebraic relationship between variables.
• Vertical Line Test: A curve is the graph of a function if and only if no vertical line intersects it more than once. This validates the “unique output” requirement.
• Domain and Range Visibility: The domain is the projection of the graph onto the $x$-axis, and the range is the projection onto the $y$-axis.
• Intercepts: Points where the graph crosses the axes ($x$-intercepts where $f(x)=0$ and $y$-intercept where $x=0$).

• How to read: “The x intercepts occur where f of x equals zero, and the y intercept occurs where x equals zero.”
• Meaning: Zeros of $f$ hit the $x$-axis; evaluating $f(0)$ gives the $y$-intercept (if $0$ is in the domain).

Connected Concepts

• Graphs of Functions: Properties
• Polynomial Functions, Polynomial Graphs
• Graph Intercepts, Graph Symmetry