Lesson Video: Increasing and Decreasing Intervals of a Function Mathematics • Second Year of Secondary School

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.