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

Video Transcript

Consider the graph of the function 𝑓 of 𝑥 is equal to one divided by one minus 𝑥 plus two. What is the end behavior of the graph as 𝑥 approaches one? Option (A) the value of 𝑦 approaches ∞ when 𝑥 gets closer to one from the positive direction and approaches negative ∞ when 𝑥 gets closer to one from the negative direction. Option (B) the value of 𝑦 approaches negative ∞ when 𝑥 gets closer to one from the positive direction and approaches ∞ when 𝑥 gets closer to one from the negative direction. (C) The value of 𝑦 approaches ∞ when 𝑥 gets closer to one from the negative direction or from the positive direction. Or is it option (D) the value of 𝑦 approaches negative ∞ when 𝑥 gets closer to one from the negative direction or the positive direction?

In this question, we’re given the graph of a function: 𝑓 of 𝑥 is equal to one divided by one minus 𝑥 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 𝑥 approach one. And there’s many different ways we could go about answering this question. For example, we can determine the end behavior of this graph as 𝑥 approaches one entirely by using the given function.

To see how we might do this, let’s consider the denominator in the given function: one minus 𝑥. If our values of 𝑥 are greater than one, which means 𝑥 is in the positive direction of one, then we can note one minus 𝑥 will be negative, since we’re subtracting a number bigger than one from one. We can also note as the values of 𝑥 get closer and closer to one, the magnitude of one minus 𝑥 will get closer and closer to zero. In other words, the distance between one and 𝑥 gets smaller as 𝑥 gets closer and closer to one. So we’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 won’t change this fact. So 𝑓 of 𝑥 will approach negative ∞ as our values of 𝑥 approach one from positive direction.

However, this level of analysis is very difficult. So, instead, it’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 𝑥-coordinates of the point on the curve and the output values are the corresponding 𝑦-coordinates. This means we can determine what happens to the outputs of the function as our values of 𝑥 approach one from the positive direction by seeing what happens to the 𝑦-coordinates of points which lie on the curve.

To do this, let’s add in the vertical line 𝑥 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 𝑥 approach one from the right, the 𝑦-coordinates of the points on the curve are going lower and lower. They’re unbounded, so they’re approaching negative ∞. And we get a similar story if we look at the values of 𝑥 approaching one from the negative direction. We can see that the 𝑦-coordinates of the point on the curve are getting larger and larger, and they’re unbounded. So they’re approaching positive ∞.

And we can see that this only matches option (B). The value of 𝑦 approaches negative ∞ when 𝑥 gets closer to one from the positive direction and approaches ∞ when 𝑥 gets closer to one from the negative direction.