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.