Figure 10-1: Graph of function f(x) = x3 - x
A two dimensional graph is a diagram of a numerical relation in which any ordered pair in the relation corresponds to a point on a Cartesian plane. A function with one independent variable defines a rule for determining ordered pairs. By convention, the independent variable is represented on this graph as displacement from the origin on the x-axis or ordinate. The dependent variable is the displacement from the origin on the y-axis or abscissa. For example, consider the graph:
Figure 10-1: Graph of function f(x) = x3 - x
This function has a domain and a codomain that are in the set of real numbers as well as a range that is the set of real numbers. It is defined as:
f: R -> R, f(x) = x3 - x
It is possible to visualize the domain and range of a function using its graph. The domain is the projection of the graph onto the x-axis. The range is the projection of the graph onto the y-axis. Above, it is obvious that both the domain and range will be the entire x-axis and y-axis respectively. Consider this graph:
Figure 10-2: Graph of function y = √x
The projection of the graph onto the x-axis is the interval [0, +∞); the projection onto the y-axis is the interval [0, +∞). These intervals are the domain and range of the function y = √x.
If a line can be drawn parallel to the y-axis that cuts a graph at most once, the graph is of a function. If such a line may cut the graph more than once, the graph cannot be of a function.
Since a function returns one value for a valid variable, a vertical line defined by x = c, would cross its graph at most once. For the graphs above, we can see that for any point in the domain of the function there is only one point in the range. For any point on the range of a function there can be one or more points on the graph.
If a graph is of a function there is at most one value of y for any value of x. The horizontal line test implies a one-to-one function, meaning that there is only one value of x associated with each value of y.