MA005: Calculus I

The definition of function requires that each element of the domain, each input value, be sent by the function to exactly one element of the range, to exactly one output value, so for each input x-value there will be exactly one output y–value, $y = f(x)$. The points $(x, y_1)$ and $(x,y_2)$ cannot both be on the graph of $f$ unless $y_1 = y_2$. The graphic interpretation of this result is called the Vertical Line Test.

Vertical Line Test for a Function:

A graph is the graph of a function if and only if a vertical line drawn through any point in the domain intersects the graph at exactly one point. 

Fig. 6(a) shows the graph of a function. Figs. 6(b) and 6(c) show graphs which are not the graphs of functions, and vertical lines are shown which intersect those graphs at more than one point. Non–functions are not "bad", and sometimes they are necessary to describe complicated phenomena.