Question Video: Discussing the Monotonicity of a Graphed Function Mathematics

Which of the following statements correctly describe the monotony of the function represented in the figure below? [A] The function is increasing on (5, 8), constant on (−1, 5), and decreasing on (−2, −1). [B] The function is increasing on (−2, −1), constant on (−1, 5), and decreasing on (5, 8). [C] The function is increasing on (5, 8) and decreasing on (−2, 5). [D] The function is increasing on (−2, 5) and decreasing on (5, 8).

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Video Transcript

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.

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