In order to discuss functions, you need to understand their basic characteristics.

A function describes a relationship between one or more inputs and one output. The inputs to a function are variables; x is the most common variable in the functions that appear on the Math IIC, though you may also come across an occasional a, b, or some other letter. The output of the function for a particular value of x is usually represented as f(x) or g(x). When a function of a single variable is graphed on the x-y plane, the output of the function, f(x), is graphed on the y-axis; functions are therefore commonly written as y = x² rather then f(x) = x².

Two characteristics of functions that you should become comfortable with are domain and range. The domain is the set of inputs (x values) for which the function is defined. Consider the following two functions: f(x) = x² and g(x) = ¹/_x. In f(x), any value of x can produce a valid result since any number can be squared. In g(x), though, not every value of x can generate an output: when x = 0, g(x) is undefined.

The range of a function is closely related to the domain. Whereas the domain is the set of inputs that a function can take, the range is the set of outputs that a function can produce. To help you understand this concept, let’s use the examples in the last paragraph: f(x) = x² and g(x) = ¹/_x. Try to think of all the values that can be generated when a number is squared. Well, all squares are positive (or equal to 0), so f(x) can never be negative. In the case of g(x), almost every number is part of the range. In fact, the only number that cannot be generated by the function g(x) is 0. Try it for yourself; there’s no value of x for which ¹/_x equals 0. The range of the function g(x) is all numbers except zero.

Once you understand the concepts of a function’s domain and range, you can see how their relationship helps to define a function. A function requires that each value of x only has one value of f(x); that is, each element of the domain must be paired with exactly one element of the range. Each element of the domain and its corresponding element of the range can be written (and graphed) as a coordinate pair, (x, f(x)).

Now consider the set of coordinates {(1, 5), (3, 5), (1, 3)}. Does this set define a function? No, because the definition of a function requires that each element of the domain is paired with only one element of the range. Specifically, 1 has been assigned to two different values in the range, 5 and 3. This rule is easy to apply when you have the coordinates listed for you. If you are presented with a graph instead, you can use the “vertical line test,” which states that any vertical line drawn anywhere along the function must not intersect the would-be function more than once.