1.4 The Domain and Range of a Function
Consider the
function 
.
Notice that
there is no output number when the input number equals 2, since division by
zero is meaningless. Thus x can
have any value except 2. We say that 2 is
not in the domain of the function f.
Here is another
example. Let 
. In this course, we
will only consider real-valued functions. This means that no output numbers may contain a multiple of 
, or the imaginary number i. Thus we cannot allow an input number such as
0 for the function g, since 
. So zero is not in
the domain of g. Furthermore, no
value of x for which x < 4 is in the domain of g. [Why
is this true?]
The implicit
domain principle: It will
be assumed that the domain of a function contains every possible input
number unless otherwise specified. Only
inputs which imply division by zero or which would produce an output containing
the square root of −1 are excluded.
For example, the
implicit domain of g is the interval [4,∞).
[Why is this true?]
However, it is
always possible to restrict the domain of a function to be something less
than the implicit domain.
For example, one
could define 
for x > 3. Then the explicit domain is the
interval (3,∞). If it were not
for the explicit restriction on the domain of p it would have an
implicit domain of (-∞,∞).
For each of the
following exercises find the implicit domain of the function. Write the domains in interval notation.
Exercise
1.4.1
Exercise
1.4.2
Exercise
1.4.3
We call the set
of all inputs of a function its domain, and we call the set of all outputs of a
function its range.
It is much
easier, in general, to look at the equation of a function and figure out its
domain than it is to figure out its range.
For example,
take 
. We can see that its
domain is all real numbers except 3. In
interval notation that is written 
. It is not as easy
to see what the the range must be. One technique which sometimes works is to replace the 
in the equation with y
and solve the equation for x.
When we do this with this example, we find that 
. Thus we see that
the output number y can be anything except 1. Thus, the range of the function is 
.
Exercise
1.4.4
Find the domain
and the range of 
. Express the answers
in interval notation.
Another technique
for finding the range is to sketch the graph and see what kind of y
values points on the graph may have.
For example, if we graph 
, we see that all the y coordinates of points on the
graph are greater than or equal to zero.
So the range is 
.
Exercise
1.4.5
Sketch the graph
of 
. Find its domain and
its range.