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 ∞.
