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.