Question Video: Describing the End Behavior of a Function from Its Graph Mathematics • 10th Grade

Consider the graph of the function ๐‘“(๐‘ฅ) = 1/(1 โˆ’ ๐‘ฅ) + 2. What is the end behavior of the graph as ๐‘ฅ approaches 1? [A] The value of ๐‘ฆ approaches โˆž when ๐‘ฅ gets closer to 1 from the positive direction and approaches โˆ’โˆž when ๐‘ฅ gets closer to 1 from the negative direction. [B] The value of ๐‘ฆ approaches โˆ’โˆž when ๐‘ฅ gets closer to 1 from the positive direction and approaches โˆž when ๐‘ฅ gets closer to 1 from the negative direction. [C] The value of ๐‘ฆ approaches โˆž when ๐‘ฅ gets closer to 1 from the negative direction or from the positive direction. [D] The value of ๐‘ฆ approaches โˆ’โˆž when ๐‘ฅ gets closer to 1 from the negative direction or the positive direction.

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Video Transcript

Consider the graph of the function ๐‘“ of ๐‘ฅ is equal to one divided by one minus ๐‘ฅ plus two. What is the end behavior of the graph as ๐‘ฅ approaches one? Option (A) the value of ๐‘ฆ approaches โˆž when ๐‘ฅ gets closer to one from the positive direction and approaches negative โˆž when ๐‘ฅ gets closer to one from the negative direction. Option (B) the value of ๐‘ฆ approaches negative โˆž when ๐‘ฅ gets closer to one from the positive direction and approaches โˆž when ๐‘ฅ gets closer to one from the negative direction. (C) The value of ๐‘ฆ approaches โˆž when ๐‘ฅ gets closer to one from the negative direction or from the positive direction. Or is it option (D) the value of ๐‘ฆ approaches negative โˆž when ๐‘ฅ gets closer to one from the negative direction or the positive direction?

In this question, weโ€™re given the graph of a function: ๐‘“ of ๐‘ฅ is equal to one divided by one minus ๐‘ฅ plus two. And we need to use this graph or the given function to determine the end behavior of this graph as the values of ๐‘ฅ approach one. And thereโ€™s many different ways we could go about answering this question. For example, we can determine the end behavior of this graph as ๐‘ฅ approaches one entirely by using the given function.

To see how we might do this, letโ€™s consider the denominator in the given function: one minus ๐‘ฅ. If our values of ๐‘ฅ are greater than one, which means ๐‘ฅ is in the positive direction of one, then we can note one minus ๐‘ฅ will be negative, since weโ€™re subtracting a number bigger than one from one. We can also note as the values of ๐‘ฅ get closer and closer to one, the magnitude of one minus ๐‘ฅ will get closer and closer to zero. In other words, the distance between one and ๐‘ฅ gets smaller as ๐‘ฅ gets closer and closer to one. So weโ€™re dividing one by a negative number with a very small magnitude. And as the magnitude of this number gets smaller, its reciprocal will get larger in magnitude. And adding two to this value wonโ€™t change this fact. So ๐‘“ of ๐‘ฅ will approach negative โˆž as our values of ๐‘ฅ approach one from positive direction.

However, this level of analysis is very difficult. So, instead, itโ€™s very useful to be able to do this from a given diagram or, if possible, to sketch a diagram to help us determine this information. We can do this by recalling the input values of the function are the ๐‘ฅ-coordinates of the point on the curve and the output values are the corresponding ๐‘ฆ-coordinates. This means we can determine what happens to the outputs of the function as our values of ๐‘ฅ approach one from the positive direction by seeing what happens to the ๐‘ฆ-coordinates of points which lie on the curve.

To do this, letโ€™s add in the vertical line ๐‘ฅ is equal to one onto our diagram. We can see that the curve gets closer and closer to this line. However, it never touches the line. This is a vertical asymptote of the curve. In particular, we can see as the values of ๐‘ฅ approach one from the right, the ๐‘ฆ-coordinates of the points on the curve are going lower and lower. Theyโ€™re unbounded, so theyโ€™re approaching negative โˆž. And we get a similar story if we look at the values of ๐‘ฅ approaching one from the negative direction. We can see that the ๐‘ฆ-coordinates of the point on the curve are getting larger and larger, and theyโ€™re unbounded. So theyโ€™re approaching positive โˆž.

And we can see that this only matches option (B). The value of ๐‘ฆ approaches negative โˆž when ๐‘ฅ gets closer to one from the positive direction and approaches โˆž when ๐‘ฅ gets closer to one from the negative direction.

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