Video: Discussing the Monotonicity of a Graphed Function

The graph of a function is given below. Which of the following statements about the function is true? [A] The function is decreasing on ℝ. [B] The function is constant on ℝ. [C] The function is increasing on (−∞, 0]. [D] The function is increasing on ℝ. [E] The function is constant on (−∞, 0].

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Video Transcript

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.

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