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.
