# Graph of a Function

The graph of a function f is defined to be the graph of the corresponding equation $y = f\left( x \right)$. In other words, the graph $f$ is the set of all points $\left( {x,y} \right)$ in the Cartesian plane such that $x$ is in the domain of $f$ and $y = f\left( x \right)$.

For instance, if m and b are constants, then the graph of the function $f\left( x \right) = mx + b$ is the same as the graph of the equation $y = mx + b$, a line with slope $m$ and $y$ intercept b (as shown in the figure below). For this reason, a function of the form $f\left( x \right) = mx + b$ is called a linear function.

Graphs of functions that are not linear are often (but not always) smooth curves in the Cartesian plane.

The Vertical-Line Test

A set of points in the Cartesian plane is the graph of a function if and only if no vertical straight line intersects the set more than once.