Video Transcript
>Which of the following represents
the function 𝑔 of 𝑥 whose asymptotes are the same as those of the graphed function
𝑓 of 𝑥? Option (A), 𝑔 of 𝑥 is equal to
one divided by 𝑥 minus two minus three. Option (B), 𝑔 of 𝑥 is equal to
one divided by 𝑥 plus two plus three. Option (C), 𝑔 of 𝑥 is equal to
one divided by 𝑥 minus two plus three. (D) 𝑔 of 𝑥 is equal to one
divided by 𝑥 minus three plus two. Or is it option (E), 𝑔 of 𝑥 is
equal to one divided by 𝑥 plus three plus two?
In this question, we’re asked to
determine which of five given functions represents a function 𝑔 of 𝑥. And to help us determine this,
we’re given a graph of a function 𝑓 of 𝑥. And we’re told that the asymptotes
of 𝑓 of 𝑥 and 𝑔 of 𝑥 are the same. So, we need to use the graph of the
function 𝑓 of 𝑥 to determine its asymptotes. And to do this, let’s start by
recalling what we mean by the asymptotes of a function. The asymptotes of the graph of a
function are straight lines which the graph approaches. In particular, horizontal
asymptotes are horizontal lines that the graph of the function approaches and
vertical asymptotes are vertical lines which the graph of the function
approaches.
And by looking at the given graph
of 𝑓 of 𝑥, we can see that it has two asymptotes. First, we can see it has a
horizontal asymptote as shown. This is because as the values of 𝑥
approach ∞, we can see that the graph of the function is approaching this horizontal
line. And we can also see the same thing
is true as our values of 𝑥 approach negative ∞. It’s approaching this horizontal
line. And this is the only horizontal
line where this is true. So, we can see there’s one
horizontal asymptote of this line, and we can see it lies between 𝑦 is equal to two
and four. Since it’s exactly halfway between
these two values, this is the horizontal line 𝑦 is equal to three.
We can follow very similar
reasoning to show the vertical line 𝑥 is equal to two is a vertical asymptote of
our function 𝑓 of 𝑥. One way of seeing this is to note
as our values of 𝑥 are approaching two from the positive direction, we can see the
outputs of the function are decreasing without bound. Similarly as the values of 𝑥
approach two from the negative direction, we can see the outputs of the function,
that’s the 𝑦-coordinates of the points on the curve, are increasing without
bound. The curve is getting closer and
closer to the line 𝑥 is equal to two.
So, the graph of the function 𝑓 of
𝑥 has two asymptotes: the horizontal asymptote 𝑦 is equal to three and the
vertical asymptote 𝑥 is equal to two. But remember, we’re told that our
function 𝑔 of 𝑥 has the same asymptotes. So, 𝑔 of 𝑥 must also have these
two asymptotes. Let’s now look at the five given
options to see which of these options also has these two asymptotes.
And there’s a few different ways we
could go about doing this. One way is to recall that we know
the graph 𝑦 is equal to one divided by 𝑥 minus 𝑎 plus 𝑏 for constants 𝑎 and 𝑏
will have two asymptotes, 𝑥 is equal to 𝑎 and 𝑦 is equal to 𝑏. This is because as our values of 𝑥
approach 𝑎, the magnitude of 𝑥 minus 𝑎 is going to get smaller and smaller. And dividing one by a term whose
magnitude is getting smaller and smaller is going to grow without bound. So, as 𝑥 approaches 𝑎, the
outputs of this function will either go to ∞ or negative ∞. 𝑥 is equal to 𝑎 is a vertical
asymptote of this curve. Similarly as the values of 𝑥
approach positive ∞ or negative ∞, the first term is going to approach zero, which
means we will have a horizontal asymptote 𝑦 is equal to 𝑏.
Therefore, we can find a function
with the same asymptotes as 𝑓 of 𝑥 by using 𝑎 is equal to two and 𝑏 is equal to
three. And it’s important to note we’re
subtracting 𝑎 in the denominator. So, we’ll get one divided by 𝑥
minus two plus three. And we can see this is option
(C). We know it has a vertical asymptote
at 𝑥 is equal to two and a horizontal asymptote at 𝑦 is equal to three.
And it’s worth noting that we can
show this is the only possible correct option. For example, the graph of the
function given in option (A) will have a horizontal asymptote when 𝑦 is negative
three. The graph of the function in option
(B) will have a vertical asymptote when 𝑥 is equal to negative two. And the graphs of the functions in
options (D) and (E) both have a horizontal asymptote when 𝑦 is equal to two. And we can also see that the graphs
of all five of the given functions will only have two asymptotes.
Therefore, of the five given
options, only option (C) 𝑔 of 𝑥 is equal to one divided by 𝑥 minus two plus three
has the same asymptotes as those given in the graphed function 𝑓 of 𝑥.
