[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.

c. How does the graph behave near x = 1 and x = −1?

Give reasons for your answers.

y = (3/2)(x − (1 / x))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 5 divided by 4 multiplied by the cube root of 2 X minus 8 divided by X to the power of 2 is shown. Determine the behavior of the graph near X equals -2, and X equals 2. Awesome. So it appears for this particular problem we're asked to determine the behavior of the graph for Y near X equals -2, and X equals 2, and that is our final answer that we're ultimately trying to solve for as we're trying to describe and determine this behavior of the graph near X equals -2 and X equals 2. So with that said, let's quickly look at our graph that is provided to us by the prom itself. And as you can see, we have what appears to be two curves represented in red that mirror each other. It's like an open book, like if you were to fold. The right curve onto the left curve, it would match it, and if you were to fold the left curve on top of the right curve, it would match its shape, and it appears to be shaped like a check mark, and it looks like both curves extend vertically upwards following the Y axis going up for infinity. So it just keeps increasing, going and stretching up and up and up for infinity. So with that said, let us recall a note based off of our knowledge that from looking at our graph, the curve has a V shape, which is our checkmark shape, near X equals -2 and X equals 2, with the function touching but not crossing these points smoothly. So we need to note that the function is continuous at these points, but has a sharp turn or an abrupt change in direction. So just to quickly note, you can see, I'll circle it in blue, that once again, Our curve has a V shape near X equals -2 and X equals 2. And we need to note once again, since this is really important to note that since the function is continuous at these points, but has a sharp turn or an abrupt change in direction, this therefore suggests that the function has cusps at X equals -2 and X equals 2. So that would mean therefore, without a doubt, our final answer would have to be the graph has a V shape or a sharp turn near X equals -2 and X equals 2, which suggests cusps at these points. And that's it. That is our final answer. 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.

c. How does the graph behave near x = 1 and x = −1?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Key Concepts

Graphing Functions

Behavior Near Specific Points

Fractional Exponents

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