To answer each question, refer to the following basic graphs. Whi...

Question

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

Accepted Answer

Hey, everyone here we are asked to determine which of the following is the graph of F of X is equal to the cubed root of five X. We are also asked to determine if an open interval at which the function is decreasing exists. So here we have four answer choice options. Each displaying a graph of the given function and a statement of whether there is or is not an open interval at which the function is decreasing. So to begin solving this problem, we first need to focus on graphing our function. And so as we recall, the given function is F of X is equal to the cubed root of five X. So to begin graphing this function, we just need to consider arbitrary values of X, we can first consider when X is equal to zero. So to find the output for when X is zero, we can just substitute in 04 X in our function. So we have F of zero is equal to the cubed root of five times zero. And now simplifying, we see we just get an output of zero. So we see that one of our 100 pairs is at zero and zero. So now we want to repeat this process by creating a table of values for X and Y. So then that we can graph these ordered pairs and thus graph our function. So we can choose arbitrary values of X. We can start, let's say at negative two, then negative one, we can also include zero and then positive one and positive two. So now we just need to substitute each of these X values into our function to get our Y values. And we end up with negative 2.15, negative 1.710, positive 1.71 and positive 2.15. So now we see here that we have five different ordered pairs. And so the next step is to graph each of these points. And so now we have a graph with five of our points and one of these points is at the origin. And so with these points graphed, the next step is to complete our graph of our function by creating or drawing in a smooth line that connects each of these points. So doing this, we end up with our graph of our given function. So we now have the function F of X graphed and it crosses through each of our points that we came up with. And now looking at our graph of our given function, we see that there are or there is no interval at which the function is decreasing. Because at any interval, we see that the function is in fact increasing. Or in other words, we see that F of X sub one is less than F of X sub two for when we have that X sub one is less than X of two. So again, we see that there is no interval for which the function is decreasing. And so with this piece of information, as well as our graph of our given function, we see that we are left here with answer choice A where we have our smooth curve to represent our function F of X. And we also have that there is no open interval at which the function is decreasing. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=∛x? Is there any open interval over which the function is decreasing?

Key Concepts

Graph of the Cube Root Function

Increasing and Decreasing Functions

Open Intervals

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