1.3 Graphs of Functions
In your first
course in algebra, you learned to graph lines, and possibly other types of
graphs, such as parabolas and circles.
Consider the
following two equations:
Equation 1: 
Equation 2: 
Exercise
1.3.1
Explain why, in
the first equation, y is a function of x, and why, in the second
equation y is not a function of x.
Exercise
1.3.2
Sketch the
graphs of the two equations.
One may use the
graph of a function to estimate the output for any given input.
Look at your
graph of equation 1 and select some number on the x-axis as the input
into the function 
. Label the input
number x.
Now draw a
vertical line through the number you selected.
At the point
where the vertical line intersects the graph, draw a horizontal line.
Look at where
the horizontal line crosses the y-axis.
The number where the horizontal line crosses the y-axis is the
output corresponding to the input you selected on the x-axis. Label the output number y.
The graph of
every function operates in this way.
Now try to apply
the same process to the graph of the second equation 
, which, remember, is not a function. Notice that most vertical lines intersect
the graph in more than one place. Thus,
one must draw two horizontal lines that intersect the graph in two
places. So this cannot be the graph of
a function, since a function has only one output for any given input.
These two
examples illustrate an important principle:
No vertical
line intersects the graph of a function more than once.
This principle
is sometimes referred to as the vertical line test.