__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.