Video: Identifying the Increasing and Decreasing Regions of a Graph

The graph of a function is given below. Which of the following statements about the function is true? [A] The function is increasing on (−∞, 0) and increasing on (0, ∞). [B] The function is decreasing on (−∞, −5) and decreasing on (−5, ∞). [C] The function is increasing on (−∞, −5) and increasing on (−5, ∞). [D] The function is decreasing on (−∞, 0) and decreasing 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 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 ∞.

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