The domain
of a function is the set of all possible input values. If the domain
is not specified, we usually take it to be the set of all real numbers
for which the function is defined. The range of a function is
the set of all outputs or in other words, the set of all possible answers
when the domain values are substituted for the input variable. When
a function is given by a formula the range is often difficult to determine.
If we can graph the function, we can look at its graph to determine
the range.
Example
A Function Given by a Table of Values
The following
table gives U.S. population in millions in the indicated year:
Year

1960

1970

1980

1990

U.S.
Population (in millions)

181

205

228

250

Source: Statistical
Abstracts of the United States,1993. We can think of population as depending
on time in years so the independent variable or input is the year and
the dependent variable or output is the U. S. population. Since
the table gives a unique population for each year, it represents a function.
The domain is the set of years {1960, 1970, 1980, 1990} and the range
is the set of populations in those years {181 million, 205 million,
228 million, 250 million}.
Example A
Rational Function
Determine
the domain of the function
_{}.
Solution
The function
g(x) is not defined when x = 3 since division by 0 is not allowed.
The only problem in evaluating g(x) occurs when the denominator equals
0, that is, when x = 3, so the domain of g is the set of all real numbers
except 3. In set notation we write
_{}
or in interval notation
_{}.
In general, the domain of
a rational function is the set of all numbers for which the denominator
is not zero. Since _{}, the range is the set of all real numbers
except 0.
Example
The Square Root Function
A different
kind of situation is presented by the square root function
_{} No negative numbers can be substituted for x because
you can't take the square root of a negative number and get a real number.
But anything else is O.K. to substitute for x, so the domain of h is
the set of all nonnegative real numbers,
_{}or _{}.
Since _{} for all x, the range is also the
set of nonnegative real numbers.
Example
Linear Functions
Determine
the domain and range of the linear function
_{}.
Solution
Since any real number can be
substituted for x, the domain consists of all real numbers. A graph
of the function in Figure 1 below shows that the range is also the set
of all real numbers. Figure 1
Example
Determine the domain and range
of the quadratic function
_{}
Solution Again, since any real number
can be substituted for x, the domain consists of all real numbers. Since
the graph of this function is a parabola opening up, it has a minimum
value, which is the ycoordinate of the vertex. Using the vertex
formula we find that the vertex is
_{}
and
_{}.
So, the range is
_{}.
See Figure
2. Figure 2
Example Polynomial
Functions
The domain
of a polynomial function is always the set of all real numbers. However,
the range depends on the particular function, so you should always graph
the function to determine the range. As an example, consider the cubic
function
_{}
whose
graph is shown in Figure 3.
Figure 3
Example The
Sine and Cosine Functions
Recall that
the sine and cosine are functions of a real number _{}, the radian measure of an angle. Also, given any
real number _{}, there is an angle with measure _{} radians. So, the domain of these two trigonometric
functions is the set of all real numbers. Since the maximum output
value is 1 and the minimum output value is –1, the range is the interval
[1, 1].
Example
A Function Representing a RealWorld Situation
There are practical considerations
to take into account when the function represents a real problem. Consider
the function J(p) = the unemployment rate (percentage of people unemployed)
when the gross national product is p.
Solution
The input
is the gross national product, the monetary value of all goods produced
in the country. Thus p ³ 0 and the domain of function
J is all nonnegative real numbers,
_{}. The range of the function is the set
of real all numbers between 0 and 100 since the unemployment rate is
a percentage.

