[Technology Exercise] Graph the functions in Exercises 113 and...

Question

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

b. How does the graph behave as x → ±∞?

Give reasons for your answers.

y = (3/2)(x / (x − 1))²/³

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The graph of Y equals 2 multiplied by the cube root of X divided by X + 1 to the power of 2 is shown. Determine the behavior of the graph as X approaches plus or minus infinity. Awesome. So it appears for this particular problem, we are given a graph for Y, and we're asked to determine, using our graph of Y, we need to determine the behavior of the graph as X approaches positive and negative infinity. So with that said, now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is as x approaches plus or minus infinity. Y approaches 2. B is as x approaches plus or minus infinity. Y approaches negative 2. C is as x approaches plus or minus infinity. Y approaches infinity. And finally, D is as x approaches plus or minus infinity, Y approaches -1. Awesome. So really quickly, let's look over our graph of why that is provided to us, and as you can see, we have a curve represented in red. And we also have a horizontal asymptote represented by a green dotted line for Y equals 2, and we have a vertical asymptote represented by a blue dotted line for X equals -1. And as you can see, our curves basically approach our asymptotes, but they do not intersect or cross our asymptotes. They get very, very close, but they never touch. Except for the one exception. Where the curve crosses through our horizontal asymptote of Y equals 2. Awesome. So with that said, and it also looks like our curve dips down and touches our origin point at 0.0 as well. So, with that said, now that we have a visualization of what's happening for this particular prong, we need to note and recall that when we look at our graph, we can see that the curve approaches the horizontal asymptote Y equals 2, as X approaches plus or minus infinity. So once again, let us note that as Y equals 2, that means that X approaches plus or minus infinity. So that will mean, without a doubt, our final answer has to be multiple choice answer letter A, which states that as X approaches plus or minus infinity, Y will approach 2. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

b. How does the graph behave as x → ±∞?

Give reasons for your answers.

y = (3/2)(x / (x − 1))²/³

Key Concepts

End Behavior of Functions

Rational Functions

Asymptotic Analysis

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