Graph y = sin x and y = ⌊sin x⌋ together. What are the domain ...

Question

Graph y = sin x and y = ⌊sin x⌋ together. What are the domain and range of ⌊sin x⌋?

Accepted Answer

Hello there. Today we're going to 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. Graph the functions Y equals cosine of X and Y equals the floor function of cosine of X on the same coordinate plane. Consider only 1 period when graphing. Determine the domain in range of y equals the floor. Function of cosine of X. Fantastic. So first off, let us quickly focus on these L-shaped brackets. These L-shaped brackets are just a fancy way of denoting the floor function or the floor equation. So that's what that's important to note. And the floor function just means that's what the floor function means, it's just anything that's within these L-shaped brackets just denotes a floor function, and the floor function, which returns the greatest integer less than or equal to the input value. So that's what these L-shaped brackets mean. So now that we understand what the floor function is, let's quickly Confirm what exactly we're trying to solve for. So first off, we're asked to graph both of these functions for Y. So we're trying to graph Y equals cosine of X, and we're also trying to graph Y equals the floor function of cosine of X on the same coordinate planes, and we're graphing them on the same graph. We're also told to consider only one period when graphing, and we're also asked to determine the domain and range of our floor function for cosine of X. So, with that said, first off, let us focus our attention on solving for Y equals cosine of X first. So Y equals cosine of X. So let's focus our attention on the first function or the first graph. So, moving right along here, so this function we need to know is in the form of Y equals A multiplied by cosine of B X minus C plus D. So, now that we can note that this functions in this particular form, it's important to note, we need to note that where the amplitude, which is denoted by capital A, so the absolute value of the amplitude is going to be equal to the absolute value of 1, which is going to be equal to one, because anything within the absolute value symbol is going to be simply equal to one. Let us also note that the period Which we're gonna denote as P, so the period P is going to be equal to 2 pi divided by B, which in this case is going to be equal to 2 pi divided by 1, which is going to be simply equal to 2 pi. And the phase shift. Which is going to be equal to C divided by B, is going to be equal to 0 divided by 2, which is going to be equal to 0. And once again, we're just following our formula or the formula form for Y equals A multiplied by a cosine of B X minus C plus D because that's the form our current function Y equals cosine of X is in. We also know the vertical translation, which we'll call D and D, in this case is simply going to be equal to 0. Now we need to divide our period into 4 equal parts, so we need to take P divided by 4. So in this case we need to take 2 pi divided by 4, and when we perform this calculation, we'll simply get pi divided by 2. Fantastic. Now we need to solve for our different values of X. So let us note that X1 is going to be simply going to be equal to 0. So once again we need to note that X1 is going to be equal to 0, and we all, we also need to solve for more x values. We need to solve for X2 X3, X4, and X5. Fantastic. So let us know that X2 is going to be equal to the value of X1 plus pi divided by 2, which when we plug in our known variables and solved, we will find that X2 is equal to pi divided by 2. X3 is going to be equal to the value of X2 plus pi divided by 2, which when we plug in our known variables, we'll get an answer of pi. X4 is going to be equal to the value of X3 plus pi divided by 2, which when we perform that calculation, we plug in all of our known variables, we'll get. 3 pi divided by 2, and finally X5 is going to be equal to the value of X4 plus pi divided by 2, which is going to be equal to when we plug in our known variables and so 2 pi. So now we need to solve for or we need to evaluate our So we're going to be evaluating Y equals cosine of X at the specific X values, so we need to solve for Y1, Y2, Y3, Y4, and Y5. Fantastic. So, moving right along here, so Y1 is going to be equal to Y of X1. Which is going to be equal to cosine of 0, which is going to be simply equal to 1. So instead of rewriting this whole expression out every single time, I'm just going to give you the values for Y2, Y3, Y4, and Y5, knowing that we're plugging in our different values for X1. So Y2 we're plugging in X2, Y3 X3, Y4 Y and so Y4 is going to be X4, and then Y5 is going to be the value for X5, and we're plugging in these different values of X into our original function. So when we do that. Y2 is going to be simply equal to 0. Y3 is going to be equal to -1, Y4 is simply gonna be 0, and Y5 is going to be equal to 1. Fantastic. So now, before we start plotting our different points, which let us quickly note that our points that we need to plot are going to be 0.1, we also are going to need to plot I divided by 2.0. We're also gonna need to plot pi-1 and 3 divided by 2.0, and we're also going to need to plot 21. And we're going to need to plot these points to connect them with a smooth curve, and we're going to pause really quick because we're going to graph everything all at the same time, but these are our plots that we're going to need to plot. Once again, these are our points to help us plot our graph of Y equals cosine of X. So that's how to plot that particular function. So now, let's take a moment here to quickly consider the floor function. So let us quickly recall and note that the floor function takes a real number X and returns the greatest integers less than or equal to X, and we need to use our graph of Y equals cosine of X and and we need to also use the definition, as we should recall of the floor function to graph. And once again, we're now considering. Y equals the floor function of cosine of X. And once again, we're using those L-shaped brackets. Awesome. So with that said, we now need to note that at X equals 0. Cosine of 0 equals 1, so thus, Y equals the floor function of cosine of 0 is going to be equal to the floor function of 1, which is going to be equal to simply 1. So for 0 is less than X, X is less than pi divided by 2. And 0 is less than cosine. Of X. So 0 is less than cosine of X, which is less than 1. So that will mean therefore that Y is equal to 0, and that X equals pi divided by 2, so we need to take pi divided by 2. So when X is equal to pi divided by 2, cosine of. Pi divided by 2, so we need to take cosine of pi divided by 2, and we're gonna say that cosine of pi divided by 2 is gonna be equal to 0. So therefore, so we need to note that thus as a result, Y is going to be equal to the floor function of cosine of pi divided by 2. Which is gonna be equal to the floor function of 0, which is going to be simply equal to 0, and for I divided by 2 is less than X and X is less than 3 pi divided by 2, that will mean that -1 is less than cosine, of X is less than 0, so thus, Y is going to be equal to -1. So We need to note that at X equals 3 pi divided by 2, so cosine of 3 pi divided by 2 is going to be equal to 0. So thus Y is going to be equal to the floor function of cosine. Of 3 I divided by 2, which is going to be equal to the floor function of 0, which is also going to be equal to 0. And finally, let's Move up to the, let's move up a little bit. Let us also note that at. X equals 2 pi. That will mean that cosine of 2 pi is gonna be equal to 1. So thus, Y is going to be equal to the floor function of cosine of 2 pi. Don't forget your brackets, so the. Your L brackets, so Y equals the floor function of cosine of 2 pi, which is going to be equal to the floor function of 1, and the floor function of 1 is simply going to be equal to 1. So now we need to graph our function of Y equals the floor function of cosine of X based on the information above and also plot our function for Y equals cosine of X. So when we do that, I've gone ahead and used graphing software to plot our points. And as you can see, we should produce the following graph. So for Y equals cosine of X, we have a blue curve with our different coordinate points plotted in yellow, and then we also have our function for Y equals the floor function of cosine of X represented in orange. And as you can see, We have the domain of Y equals the floor function of cosine of X is in the is the same of that of cosine of X, which is all real numbers. However, the range of Y equals the floor function of cosine of X will consist of integer values that the floor function of cosine can take. So since cosine of X varies between -1 and 1, which is inclusive of -1 and 1, that will mean that the range is curly bracket. Y Y equals -101. So Y equals -1.01. And that will also mean, without a doubt that the domain is going to be equal to curly bracket of X X is an element of all real numbers, and once again it's in curly brackets. And that's it. That is our final answer, or our final answers, I should say. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graph y = sin x and y = ⌊sin x⌋ together. What are the domain and range of ⌊sin x⌋?

Key Concepts

Sine Function

Floor Function

Domain and Range

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