Video Transcript
Consider the graph of the function
𝑓 of 𝑥 is equal to one divided by 𝑥 plus two. What happens to the function when
the value of 𝑥 approaches negative two? Option (A) the value of 𝑦
approaches ∞ when 𝑥 gets closer to negative two from the positive direction and
approaches negative ∞ when 𝑥 gets closer to negative two from the negative
direction. Option (B) the value of 𝑦
approaches ∞ when 𝑥 gets closer to negative two from the negative direction or from
the positive direction. (C) The value of 𝑦 approaches
negative ∞ when 𝑥 gets closer to negative two 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 negative two from the positive
direction and approaches ∞ when 𝑥 gets closer to negative two from the negative
direction?
In this question, we’re given the
graph of a function 𝑓 of 𝑥 is equal to one divided by 𝑥 plus two. And since this function is the
quotient of two polynomials, we can say 𝑓 of 𝑥 is a rational function. We want to determine what happens
to the graph of our function as our values of 𝑥 approach negative two. And there’s a few different ways we
could go about this. For example, we could determine
what happens to the output of our functions around 𝑥 is equal to negative two
directly by using the given function. For example, we could construct a
function table with our values of 𝑥 getting closer and closer to negative two.
However, this is not necessary
because we’re given a graph of the function. And even if we weren’t given a
graph of the function, we could just sketch this graph by noting it’s a translation
of the graph of one over 𝑥 two units to the left. And in the graph of a function, the
𝑥-coordinate of any point on the curve tells us the input value and the
corresponding 𝑦-coordinate of this point tells us the output value of the
function. So we can determine what happens to
the output of this function as the values of 𝑥 approach negative two by seeing what
happens to the 𝑦-coordinates of points on the curve as the input values of 𝑥
approach negative two from either direction.
To do this, let’s start by
sketching the vertical line 𝑥 is equal to negative two onto the given diagram. We can see that the curve
approaches this line. So this is a vertical asymptote of
the function. Let’s now see what happens to the
outputs of the function as 𝑥 approaches negative two from either side. Let’s start with the positive
direction. And remember this is the side from
the positive values of 𝑥. As our values of 𝑥 approach
negative two, we can see that the output values, that’s the 𝑦-coordinates of the
points on the curve, are getting larger and larger. We can in fact see that the
𝑦-coordinates are growing without bound. So, as our values of 𝑥 get closer
to negative two from the positive direction, the 𝑦-values are approaching ∞.
We can do the exact same thing to
determine what happens to the outputs of the function as the values of 𝑥 get closer
to negative two from the negative direction. This time, the 𝑦-coordinates of
the points on the curve are decreasing without bound. So they’re approaching negative
∞.
And of the four given options, we
can see that this only matches option (A). The value of 𝑦 approaches ∞ when
𝑥 gets closer to negative two from the positive direction and approaches negative ∞
when 𝑥 gets closer to negative two from the negative direction.
