Now, when finding the domain, we must make sure that our
are still in BOTH the domain of f and the domain of g. We must also
make sure that g(x) ≠ 0. So, we still have
as found in part (a),
and we now have the additional restriction on our domain that
can’t be equal to 0. This means that x can not be equal to zero . Then,
the domain for this function is , or simply x
≠ 0, and .
We are again dividing functions, this time dividing g by f. In this case,
the resulting function .
To find the domain, we still need to have x values that are in both the
domain of f and the domain of g, so is still
a requirement. Now,
we have to make sure that f(x) ≠ 0, so 2x + 3 can’t be 0. This means
that , and the domain is
4. Use the graph of the above function to answer the
(a) Is f(x) even, odd, or neither?
This is an even function. We can tell by noticing that the graph
is symmetric about the y-axis. If you were to fold the page along
the x-axis, you’d notice that both halves of the graph lay perfectly
on top of one another . Alternatively, try to find some point x for
which f(x) and f(−x) are different. You could search forever, and
you’d never find one.
(b) Find f(1).
In order to find the value of f(1), we look for the y value when
x = 1. Since this y value is −1, we know that f(1) = −1.
(c) On which intervals is f(x) increasing?
As we move from left to right starting at x = −1, we notice that
the y values increase as the x values increase until we get to x = 0.
Therefore, we say that f(x) is increasing on the interval (−1, 0).
This also happens as we move from x = 1 to x = 1.5. So, the
function is also increasing on the interval (1, 1.5). Remember, we
never include the endpoints when talking about intervals on which
a function is increasing.
(d) What is the domain of f(x)?
There are no arrows on the graph to indicate that it continues
beyond the picture that we have on the page, so if we are look-
ing for the domain, we do not include any x values that are not
physically on the graph. Since there are no breaks in the graph or
open circles, we know that all x values shown are actually part of
the domain. Therefore, we can say that the domain is all values
of x such that −1.5 ≤ x ≤ 1.5. In interval notation, you would
write that the domain is [−1.5, 1.5].
(e) What is the range?
Again, there is nothing to indicate that the graph continues, so
we only include values based on the information given. Therefore,
since the y values on the graph range from −1 to 1 with no breaks,
we can say that the range of f(x) is all y values such that
−1≤ y ≤1. In interval notation, this is written as [−1, 1].
Notice that all points close to the point (0, 1) have lower y val-
ues. So, at x = 1, the function has a local maximum. The local
maximum value is 1.
(g) For which x values does f(x) = −1?
This is the same as asking for the x coordinates of any points on
the graph that have a y value of −1. Since the graph includes the
points (−1,−1) and (1,−1), we know that the x values for which
f(x) = −1 are x = −1 and x = 1.
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