


Characteristics of a Function
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 xy plane, the
output of the function, f(x),
is graphed on the yaxis; functions are therefore
commonly written as y = x^{2} rather
then f(x) = x^{2}.
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)
= x^{2} 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) = x^{2} 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 wouldbe function more than once.
