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Question Video: Identifying Functions with the Same Asymptotes Mathematics • 10th Grade

Which of the following represents the function 𝑔(π‘₯) whose asymptotes are the same as those of the graphed function 𝑓(π‘₯)? [A] 𝑔(π‘₯) = (1/(π‘₯ βˆ’ 2)) βˆ’ 3 [B] 𝑔(π‘₯) = (1/(π‘₯ + 2)) + 3 [C] 𝑔(π‘₯) = (1/(π‘₯ βˆ’ 2)) + 3 [D] 𝑔(π‘₯) = (1/(π‘₯ βˆ’ 3)) + 2 [E] 𝑔(π‘₯) = (1/(π‘₯ + 3)) + 2

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Video Transcript

>Which of the following represents the function 𝑔 of π‘₯ whose asymptotes are the same as those of the graphed function 𝑓 of π‘₯? Option (A), 𝑔 of π‘₯ is equal to one divided by π‘₯ minus two minus three. Option (B), 𝑔 of π‘₯ is equal to one divided by π‘₯ plus two plus three. Option (C), 𝑔 of π‘₯ is equal to one divided by π‘₯ minus two plus three. (D) 𝑔 of π‘₯ is equal to one divided by π‘₯ minus three plus two. Or is it option (E), 𝑔 of π‘₯ is equal to one divided by π‘₯ plus three plus two?

In this question, we’re asked to determine which of five given functions represents a function 𝑔 of π‘₯. And to help us determine this, we’re given a graph of a function 𝑓 of π‘₯. And we’re told that the asymptotes of 𝑓 of π‘₯ and 𝑔 of π‘₯ are the same. So, we need to use the graph of the function 𝑓 of π‘₯ 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 π‘₯, 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 π‘₯ 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 π‘₯ 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 π‘₯ is equal to two is a vertical asymptote of our function 𝑓 of π‘₯. One way of seeing this is to note as our values of π‘₯ are approaching two from the positive direction, we can see the outputs of the function are decreasing without bound. Similarly as the values of π‘₯ 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 π‘₯ is equal to two.

So, the graph of the function 𝑓 of π‘₯ has two asymptotes: the horizontal asymptote 𝑦 is equal to three and the vertical asymptote π‘₯ is equal to two. But remember, we’re told that our function 𝑔 of π‘₯ has the same asymptotes. So, 𝑔 of π‘₯ 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 π‘₯ minus π‘Ž plus 𝑏 for constants π‘Ž and 𝑏 will have two asymptotes, π‘₯ is equal to π‘Ž and 𝑦 is equal to 𝑏. This is because as our values of π‘₯ approach π‘Ž, the magnitude of π‘₯ minus π‘Ž 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 π‘₯ approaches π‘Ž, the outputs of this function will either go to ∞ or negative ∞. π‘₯ is equal to π‘Ž is a vertical asymptote of this curve. Similarly as the values of π‘₯ 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 π‘₯ by using π‘Ž is equal to two and 𝑏 is equal to three. And it’s important to note we’re subtracting π‘Ž in the denominator. So, we’ll get one divided by π‘₯ minus two plus three. And we can see this is option (C). We know it has a vertical asymptote at π‘₯ 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 π‘₯ 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 π‘₯ is equal to one divided by π‘₯ minus two plus three has the same asymptotes as those given in the graphed function 𝑓 of π‘₯.

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