College Algebra 9th txtbk

164 CHAPTER 3 FUNCTIONS
MATCHED PROBLEM 1
Determine whether each set defines a function. If it does, then state the domain and
range.
(A) S 5(2, 1), (1, 2), (0, 0), (1, 1), (2, 2)6
(B) T 5(2, 1), (1, 2), (0, 0), (1, 2), (2, 1)6
Z Defining Functions by Equations
So far, we have described a particular function in various ways: (1) by a verbal description,
(2) by a table, and (3) by a set of ordered pairs. We will see that if the domain and range are
sets of numbers, we can also define a function by an equation, or by a graph.
If the domain of a function is a large or infinite set, it may be impractical or impossible
to actually list all of the ordered pairs that belong to the function, or to display the
function in a table. Such a function can often be defined by a verbal description of the
“rule of correspondence” that clearly specifies the element of the range that corresponds
to each element of the domain. One example is “to each real number corresponds its
square.” When the domain and range are sets of numbers, the algebraic and graphical
analogs of the verbal description are the equation and graph, respectively. We will find it
valuable to be able to view a particular function from multiple perspectives—algebraic (in
terms of an equation), graphical (in terms of a graph), and numeric (in terms of a table or
ordered pairs).
Both versions of our definition of function are very general. The objects in the
domain and range can be pretty much anything, and there is no restriction on the number
of elements in each.
In this text, we are primarily interested, however, in functions with real number domains
and ranges. Unless otherwise indicated, the domain and range of a function will be sets
of real numbers. For such a function we often use an equation with two variables to specify
both the rule of correspondence and the set of ordered pairs.
Consider the equation
y x 2 2x
x any real number (1)
This equation assigns to each domain value x exactly one range value y. For example,
If x 4,
If x 1 3,
then
then
y (4) 2 2(4) 24
y ( 1 3) 2 2( 1 3) 5 9
We can view equation (1) as a function with rule of correspondence
y x 2 2x
any x corresponds to x 2 2x
The variable x is called an independent variable, indicating that values can be assigned
“independently” to x from the domain. The variable y is called a dependent variable, indicating
that the value of y “depends” on the value assigned to x and on the given equation.
In general, any variable used as a placeholder for domain values is called an independent
variable; any variable used as a placeholder for range values is called a dependent
variable.
We often refer to a value of the independent variable as the input of the function, and
the corresponding value of the dependent variable as the associated output. In this regard,
a function can be thought of as a process that accepts an input from the domain and outputs
an appropriate range element. We next address the question of which equations can be
used to define functions.