### Video Transcript

In this video, we’re going to learn
how to find the intervals over which a function is increasing, decreasing, or
constant.

We say that a function is
increasing when the value of the function 𝑓 of 𝑥 increases as the value of 𝑥
increases. This will result in a graph that
slopes upwards. And so the slope of the graph of a
function over an interval during which it is increasing must be positive. We can, conversely, say that a
function will be decreasing if the value of 𝑓 of 𝑥 decreases as the value of 𝑥
increases. It then follows that if a function
is decreasing over that interval, the slope of its graph will be negative.

For a function to be strictly
increasing or strictly decreasing, there can be no flat bits on the graph of that
function at all. If we have a flat piece of graph,
in other words, a horizontal line, we say that the function is constant over this
interval. Now, of course, we might not
necessarily be given the graph of the function, so we can generalize these
ideas. A function is increasing if when 𝑥
two is greater than 𝑥 one, 𝑓 of 𝑥 two is greater than or equal to 𝑓 of 𝑥
one. Then it’s strictly increasing if 𝑓
of 𝑥 two is just greater than 𝑓 of 𝑥 one. If when 𝑥 two is greater than 𝑥
one, 𝑓 of 𝑥 one is equal to 𝑓 of 𝑥 two, the function is constant over that
interval.

In a similar way, we form
definitions for functions that are decreasing and strictly decreasing. Now, throughout this video, we’re
also going to use interval notation to describe the intervals of increase and
decrease. So let’s recall these. 𝑅 is the set of real numbers. These are the numbers that we use
most often, and they include rational numbers and irrational numbers. But they don’t include imaginary
numbers or positive or negative ∞. Then square brackets or parentheses
describe a set of values when we do want to include the end values. And then we use the round brackets
or parentheses when we don’t want to include the end values on our interval. We’re now going to consider a
number of examples of using graphs to establish intervals of increase or decrease
and also how we’re going to find these by using the equations.

The graph of a function is given
below. Which of the following statements
about the function is true? Is it (A) the function is
decreasing on the set of real numbers? Is it (B) the function is constant
on the set of real numbers? (C) The function is increasing on
the left-open right-closed interval from negative ∞ to zero. Is it (D) the function is
increasing on the set of real numbers? Or (E) the function is constant on
the left-open right-closed interval from negative ∞ to zero.

Let’s begin by recalling what the
words decreasing, increasing, and constant tell us about the graph of a
function. If a function 𝑓 of 𝑥 is
decreasing over some interval, then the value of 𝑓 of 𝑥 decreases as the value of
𝑥 increases. In terms of the graph, we can say
that the graph will slope downwards over that interval. The opposite is true if a function
is increasing over some interval. As the value of 𝑥 increases, the
value of the function also increases. And then this looks like the graph
sloping upwards. Then if a function is constant, as
the value of 𝑥 increases, the value of the function remains the same. And in terms of the graph, this
looks like a horizontal line.

And if we compare our graph to
these three terms and these criteria, we see we have a horizontal line. So our function must be
constant. So if we compare these to our
options (A) through (E), we see we’re looking at (B) and (E). (B) says the function is constant
on the set of real numbers, whereas (E) says the function is constant on the
left-open right-closed interval from negative ∞ to zero.

So which of these are we going to
choose? If we think about this notation,
this is telling us that the function is constant for all values less than and
including zero. And in fact, this is a subset of
the set of real numbers which extends from negative ∞ to positive ∞ but doesn’t
include those endpoints. If we look at the horizontal line
representing our function, we see it has arrows at both ends. And so our line itself must also
extend up to positive ∞ and down to negative ∞. And so we can actually say that the
correct answer is (B); the function must be constant on the set of real numbers.

In our next example, we’ll see how
to use interval notation to describe whether a function is increasing, decreasing,
or constant over particular intervals.

Which of the following statements
correctly describe the monotony of the function represented in the figure below? Is it (A) the function is
increasing on the open interval five to eight, constant on the open interval
negative one to five, and decreasing on the open interval negative two to negative
one? Is it (B) the function is
increasing on the open interval negative two to negative one, constant on the open
interval negative one to five, and decreasing on the open interval five to
eight? Is it (C) the function is
increasing on the open interval five to eight and decreasing on the open interval
negative two to five? Or (D) the function is increasing
on the open interval negative two to five and decreasing on the open interval five
to eight.

So by reading the question, we’ve
probably inferred what we mean by the monotony of a function. The monotony of a function simply
tells us if the function is increasing or decreasing. And of course, we recall that if a
function is increasing over some interval, it has a positive slope. If it’s decreasing, it has a
negative slope. And if it’s constant, well, that’s
a horizontal line. So let’s look at the graph of our
function. We see it has three main
sections. The first section is between
negative two and negative one. Then the next section is between
negative one and five, whilst the third section is between five and eight.

So let’s consider each section in
turn. We can see that the slope of the
first part of our function must be positive. It’s sloping upwards. We then have a horizontal line
between 𝑥 equals negative one and five. And the third part of our graph has
a negative slope. It’s sloping downwards. Our function is therefore
increasing for sometime, it’s constant, and then finally it’s decreasing. We need to decide the intervals
over which each of these occur. It has a positive slope between 𝑥
equals negative two and negative one. And so we define this using the
open interval negative two to negative one.

We are not going to use a closed
interval. We don’t really know what’s
happening at the endpoints of this interval. For instance, when 𝑥 is equal to
negative one, the graph of our function has this sort of sharp corner. And so we’re going to leave 𝑥
equals negative two and 𝑥 equals negative one out of our interval. In a similar way, the function is
constant over the open interval negative one to five. And it’s decreasing over the open
interval five to eight. Once again, we don’t know what’s
really happening at those endpoints, but we do have sharp corners. And so we can’t say whether it’s
increasing, decreasing, or constant. And so the correct answer is (B):
the function is increasing on the open interval negative two to negative one,
constant on the open interval from negative one to five, and decreasing on the open
interval five to eight.

In our next example, we’re going to
look at how to identify increasing and decreasing regions from a reciprocal
graph.

The graph of a function is given
below. Which of the following statements
about the function is true? Is it (A) the function is
increasing on the open interval negative ∞ to zero and increasing on the open
interval zero to ∞? Is it (B) the function is
decreasing on the open interval negative ∞ to negative five and negative five to
∞? Is it (C) the function is
increasing on the open interval negative ∞ to negative five and the open interval
negative five to ∞? Or (D) the function is decreasing
on the open interval negative ∞ to zero and decreasing on the open interval zero to
∞.

Each of the statements is regarding
the monotony of the graph. It’s asking us whether the graph is
increasing or decreasing over given intervals. And so we recall that we can say
that a function is increasing if its value for 𝑓 of 𝑥 increases as the value for
𝑥 increases. In terms of the graph, we’d be
looking for a positive slope. Then if a function is decreasing,
its graph will have negative slope over that interval. And so let’s have a look at our
graph. It appears to be the graph of a
reciprocal function. And the graph has two
asymptotes. We see that the 𝑦-axis, which is
the line 𝑥 equals zero, is a vertical asymptote. And then we have a horizontal
asymptote given by the line 𝑦 equals negative five.

Now what this means is that the
graph of our function will approach these lines, but it will never quite meet
them. And this, in turn, means that the
graph of our function will never quite become a completely horizontal or completely
vertical line. And so let’s see what’s happening
as our value of 𝑥 increases. As we move from negative ∞ to zero,
the function 𝑓 of 𝑥 increases. Its slope is always positive, and
each value of 𝑓 of 𝑥 is greater than the previous value of 𝑓 of 𝑥. Then when we move from 𝑥 equals
zero to positive ∞, the same happens. And so this means that the graph is
increasing from negative ∞ to zero and from zero to ∞. But what’s happening at zero?

Well, we see that the function
can’t take a value of 𝑥 equals zero. And so the graph of our function
approaches the line 𝑥 equals zero but never quite reaches it. We then use these round brackets or
parentheses to show that the graph is increasing between 𝑥 equals negative ∞ and
zero and between 𝑥 equals zero and ∞ but that we don’t want to include the end
values in these statements. Notice that we don’t include
negative ∞ and ∞ because we can’t really define that number. And so the correct answer must be
(A), the function is increasing on the open interval negative ∞ to zero and
increasing on the open interval zero to ∞.

We’re now going to consider the
criteria for an exponential function that would make it purely increasing.

What condition must there be on 𝑧
for 𝑓 of 𝑥 equals 𝑧 over seven to the 𝑥 power, where 𝑥 is a positive number, to
be an increasing function?

For a function to be increasing, we
know that as our values for 𝑥 increase, the output 𝑓 of 𝑥 must itself also
increase. And so how can we ensure that our
function 𝑓 of 𝑥 equals 𝑧 over seven to the 𝑥 power is increasing over its entire
domain, in other words, for all values of 𝑥? Well, let’s recall what we know
about exponential functions. This is an exponential
function. And the general form of an
exponential function is 𝑓 of 𝑥 equals 𝑎 to the power of 𝑥. Now, as long as 𝑎 is positive and
a nonzero integer not equal to one, the function will be increasing if 𝑎 is greater
than one and decreasing if 𝑎 is less than one.

And so we’re going to let 𝑎 be
equal to 𝑧 over seven. And then for our function to be
increasing, 𝑧 over seven must be greater than one. This is an inequality that we can
solve just as we would solve any normal equation. We’re going to multiply both sides
by seven. 𝑧 over seven times seven is 𝑧,
and one times seven is seven. And so 𝑧 itself must be greater
than seven for the function 𝑓 of 𝑥 equals 𝑧 over seven to the 𝑥 power to be an
increasing function.

We’re now going to consider one
final example. And we’re going to look to identify
the increasing and decreasing intervals of a reciprocal function when we’ve not been
given the graph.

Which of the following statements
is true for the function ℎ of 𝑥 equals negative one over seven minus 𝑥 minus
five? Is it (A) ℎ of 𝑥 is decreasing on
the intervals negative ∞ to seven and seven to ∞? Is it (B) ℎ of 𝑥 is decreasing on
the intervals negative ∞ to negative seven and negative seven to ∞? (C) ℎ of 𝑥 is increasing on the
intervals negative ∞ to negative seven and negative seven to ∞. Or (D) ℎ of 𝑥 is increasing on the
intervals negative ∞ to seven and seven to ∞.

If we look carefully, we see that ℎ
of 𝑥 is a reciprocal function. It’s one over some polynomial. And so we know that there are
probably going to be asymptotes on our graph. Let’s think about how we might
sketch the graph of ℎ of 𝑥. We’ll begin by starting with the
function 𝑓 of 𝑥 is equal to one over 𝑥. And then we’re going to consider
the series of transformations that map the function one over 𝑥 onto the function ℎ
of 𝑥. Here is the function one over
𝑥. It has horizontal and vertical
asymptotes made up of the 𝑥- and 𝑦-axis. Now we’ll consider how we map 𝑓 of
𝑥 onto one over negative 𝑥. This is represented by reflection
in the 𝑦-axis.

And then how do we map this onto
the function one over seven minus 𝑥? Well, adding seven to the inner
part of our composite function gives us a horizontal translation by negative
seven. That’s a translation to the left
seven units. Now, in doing this, our horizontal
asymptote stays the same; it’s still the 𝑥-axis. But our vertical asymptote also
shifts left seven units. And so it goes from being the
𝑦-axis, which is the line 𝑥 equals zero, to being the line 𝑥 equals negative
seven. But of course, ℎ of 𝑥 is negative
one over seven minus 𝑥. This time, we reflect the graph in
the 𝑦-axis. And so our horizontal asymptote
remains unchanged, but our vertical asymptote is now at 𝑥 equals seven.

Our final transformation maps this
function onto ℎ of 𝑥. That’s negative one over seven
minus 𝑥 minus five. And now we move the entire graph we
translated five units down. And so we now have the graph of ℎ
of 𝑥 of negative one over seven minus 𝑥 minus five, and we’re ready to decide
whether the function is increasing or decreasing over the various intervals. Remember, if a function is
decreasing, its graph will have a negative slope, and if it’s increasing, its graph
will have a positive slope. As we move our values of 𝑥 from
left to right, that is, from negative ∞ all the way up to 𝑥 equals seven, we see
that the graph is sloping downwards. It will approach negative ∞, but
never quite reach it.

Then as 𝑥 approaches positive ∞
from seven, the graph continues to slope downwards. This time, though, it approaches
negative five. And so the function is definitely
decreasing over these intervals from negative ∞ to seven and seven to ∞. Since the function itself cannot
take a value of 𝑥 equals seven, and this is why we have the horizontal asymptote,
then we want to include open intervals. Those are the round brackets. And so the correct answer must be
(A), ℎ of 𝑥 is decreasing on the open interval negative ∞ to seven and seven to
∞.

We’ll now recap the key points from
this lesson. In this video, we learned that a
function is increasing if 𝑓 of 𝑥 increases as the value of 𝑥 increases. Over these intervals, the graph of
the function will have a positive slope or a positive gradient. And then, if the function decreases
as 𝑥 increases, we say it’s decreasing and the graph will have negative slope. Remember, we can say that a
function is strictly increasing or strictly decreasing, if there are no flat bits on
the graph at all. Finally, we saw that a function is
constant if the value of 𝑓 of 𝑥 remains unchanged as the value of 𝑥 increases and
the graph of a constant function looks like a horizontal line.