Graphing Functions
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 odd-degree 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 even-degree 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).
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