


Graphing Functions
While most of the function questions on the Math IIC will
involve analysis and manipulation of the functions themselves, you
will sometimes be asked a question about the graph of a function.
A common question of this type asks you to match a function’s graph
to its definition. The next few topics will help prepare you questions
relating to functions and their graphs.
Identifying Whether a Graph Is a Function
For the Math IIC, it is important to be able to determine
if a given graph is indeed a function. A foolproof way to do this
is to use the vertical line test: if a vertical line intersects
a graph more than once, then the graph is not a function.
The vertical line test makes sense because the definition
of a function requires that any x value have only
one y value. A vertical line has the same x value
along the entire line; if it intersects the graph more than once,
then the graph has more than one y value associated with
that x value.
The three graphs below are functions. Use the vertical
line test to prove this to yourself.
The next three graphs are not functions. In each graph,
a strategically placed vertical line will intersect the graph more
than once.
Range and Domain in Graphing
The range and domain of a function are easy enough to
see in their graphs. The domain is the set of all x values
for which the function is defined. The range is the set of all y values
for which the function is defined. To find the domain and range
of a graph, just look at which x and y values
the graph includes.
Certain kinds of graphs have specific ranges and domains
that are visible in their graphs. A line whose slope is not 0 (a
horizontal line) or undefined (a vertical line) has the set of real numbers
as its domain and range. Since a line, by definition, extends infinitely
in both directions, it passes through all possible values of x and y:
An odddegree polynomial, which is a polynomial whose
highest degree of power is an odd number, also has the set of real
numbers as its domain and range:
An evendegree polynomial, which is a polynomial whose
highest degree of power is an even number, has the set of real numbers
as its domain but has a restricted range. The range is usually bounded
at one end and unbounded at the other. The following parabola has range
{–∞, 2}:
Trigonometric functions have various domains and ranges,
depending on the function. Sine, for example, has the real numbers
for its domain and {–1, 1} for its range. A more detailed breakdown
of the domains and ranges for the various trigonometric functions
can be found in the Trigonometry chapter.
Some functions have limited domains and ranges that cannot
be simply categorized but are still obvious to see. By looking at
the graph, you can see that the function below has domain {3, ∞}
and range {–∞, –1}.
Asymptotes and Holes
There are two types of abnormalities that can further
limit the domain and range of a function: asymptotes and holes.
Being able to identify these abnormalities will help you to match
up the domain and range of a graph to its function.
An asymptote is a line that a graph approaches but never
intersects. In graphs, asymptotes are represented as dotted lines.
You’ll probably only see vertical and horizontal asymptotes on the
Math IIC, though they can exist at other slopes as well. A function
is undefined at the x value of a vertical asymptote,
thus restricting the domain of the function graphed. A function’s
range does not include the y value of a horizontal
asymptote, since the whole point of an asymptote is that the function
never actually takes on that value.
In this graph, there is a vertical asymptote at x =
1 and a horizontal asymptote at y = 1. Because
of these asymptotes, the domain of the graphed function is the set
of real numbers except 1 (x ≠ 1), and the range
of the function graphed is also the set of real numbers except 1
(f(x) ≠ 1).
A hole is an isolated point at which a function is undefined.
You’ll recognize it in a graph as an open circle at the point where
the hole occurs. Find it in the following figure:
The hole in the graph above is the point (–4, 3). This
means that the domain of the function is the set of real numbers
except 4 (x ≠ –4), and the range is the set of
real numbers except 3 (f(x) ≠
3).
