Graph each function. Give the domain and range. ƒ(x)=[[2x]]

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Hello everybody. Everything. All right. Today, today we're going to look at this question that states graph the greatest integer function given below and write the domain and range of the function. Now our function is eve is equal to the greatest integra function of three X. Now the integers provided are a domain set of a real numbers and range set of all integers. And they show us a graph with kind of dashed lines going upwards with a positive slope. B shows a domain of this set of A numbers and arrange with the set of all integers with again dash lines that if you look at, they all will move upwards. C is once again the domain with a set of our integers and arranged set of audio numbers with again dashed lines that if we look at as a whole, they are moving up the graph and D is a domain with the set of all integers and arrange a set of all real numbers with once again the lines moving. Are you fucking kidding me, dude, Jesus, man, what the fuck is up with this question? Hello everybody. Everything I today, today we're going to be on this question that states graph the greatest integer function given below and write the domain and range of the function where the function is the greatest integer function of three X. Now the answers provided all show us different types of graphs and different domains and range all graphs have in common that they show kind of dashed lines because we have a greatest integer function. We know that multiple XS will have the same Y value. So that's why they show dashed lines. And we will see that the dash lines, if we look at a pattern, they will be moving up the graph at an angle. So answer choice A shows us the domain of a set of all real numbers and arrange with the set of all integers as well as its graph. Answer choice B gives us a, a domain with the set of our real numbers and arrange with the set of all integers and a graph to go along with it. Answer choice C gives us a domain with the set of all integers and arrange with the set of all your numbers and its graph. And answer choice D gives us a domain with the set of our integers and arrange with a set of a real numbers to go along with its graph. No, for this specific question, they ask us we're dealing with the greatest in interior function. So what is the greatest interior function? Well, let's recall that the greatest integer function that can be written as open bracket, X C close bracket. So the greatest in of function is represented by brackets that will be equal to N where N is less than, or equal to X is less than N plus one. And I'll show more examples of this once we start graphing. So it's a bit more clear now to graph our function, we have to do like we would do with any other graph, we would test different points. So I always like to do a T chart when it comes to testing points where I have X values on the left and Y values on the right. So I write my T chart and for my X values, I'm going to test negative 0.9 negative 0.7, negative 0.5, negative 0.3, negative 0.1 zero, positive 0.1, positive 0.30 point 50.7 and 0.9. So I'm going to show you how to do a couple of these values and then you can try and do the rest by yourself. I'm going to test F of negative 0.9. So this is the first X that we're going to text test. So at an X of negative 0.9, we'll have F of negative 0. and that'll be equal to three multiplying negative 0.9. And once we put that into our calculator, we'll have negative 2.7. So that's what the answer. We get the answer negative 2.7 if we were just plug into our F of X, whereas three X and we plug in negative 0.9 for the X. Now we have the greatest integer function of that. So let's use the formula that we have N is less than, or equal to X is less than N plus one. So I'll have blank is less than, or equal to X, which in our case is going to be negative 2. because that's the answer we obtained from plugging in negative 0.9. And we'll have that negative 2.7 is less than blank. So now we have to find out what the end would be. So what integer is less than or equal to negative 2.7. And when you have that integer plus one, negative 2.7 is less than that. Well, in this case, that will be negative three, negative three is less than negative 2.7 and negative three plus one is negative two, which is greater than negative 2.7. So we, and when we have an X of negative 0.9, our Y will be negative three. Since we go with the N and the N, in our case here is negative three, now you can do negative 0.7 using the same technique that we just used. And once you do that, you'll see that we, we'll get another Y of negative three. And this is why in our graph, we have dashed lines, dashed horizontal lines because we have multiple X values with the same Y value. Now for an X of negative 0.5, I'll once again show you and walk you through that. So we'll have F of negative 0.5 and that is equal to three, multiplying negative 0.5. Since all we're doing is plugging it into our F O X function. And once you plug that into our calculator, we'll have that will be equal to negative 1.5. So now we can look for the N so have blank is less than, or equal to negative 1.5 is less than blank. So what integer is less than, or equal to negative 1.5? That when you add one to that integer is that the other integer would be greater than negative 1.5. Well, that is negative too negative two is less than negative 1.5 and negative two plus one is negative one, which is greater than negative 1.5. So our N would be negative two. Therefore, at an X of negative 0.5, our Y would be negative two. And now once again, you can use this same technique to continue doing the other values and I'll plug them in now while you work on them. So add an X of negative 0.3, we have a Y of negative one at an X of negative 0.1. We'll have a Y of negative one. Again, add an X of zero, we have a Y of zero and now I'll walk you through F of 0.1. I'll show you another example. So if we have F of 0.1, we'll have that equals three multiplying 0.1 simple. So we just plug it into our F of X function and that'll be equal to 0.3. So now we're using our greatest integer function, we'll have that blank is less than, or equal to 0.3 is less than blank. And now we have to look for an integer to plug in, in this case that integer is going to be zero because zero is less than 0.3 and zero plus one is one which is greater than 0.3. So our N will be zero and you can use this same technique to do the rest of the numbers that we have. The rest of the X values that we have. Once you do that, you obtain that at an X of 0.3, you'll also get a Y of zero at an X of 0.5, you'll get a Y of one at an X of 0.7, you'll get A Y of two and in an X of 0.9, you'll also get A Y of two. So now I'm just going to scroll down a bit so you have more room to work with and I'm going to write all of the points that we have obtained. So we have negative 0.9 comma negative three negative 0.7 comma negative three negative 0.5 comma negative two negative 0.3 comma negative one negative 0.1 comma negative one zero, comma zero 0.1 comma zero 0.3, comma zero 0.5, comma one 0.7 comma two N 0.9 comma two. So now we have nine points that we've obtained from trying different axes and we can use those nine points to see which graph matches with those points. Now, before we move on with that, we have to obtain a domain and range. So I'll scroll down a bit more. Not for the domain. We're dealing with the possible X values in our graph. So possible X values we can plug in any X into our function. There is nothing stopping us from plugging in an X and our graph will extend infinitely to the right and infinitely to the left. Therefore, our domain will be a set of a numbers because we can plug in any real number and we will be able to obtain a Y value. Now, for the range, it's a bit different. So for the range we're dealing with the possible Y values in our graph. So we have to see how our graph moves. We just talked about the greatest interior function will have multiple X values with the same Y because for the Y values, since we're dealing with the greatest integer, we will only have integers, there will be no decimal point in our Y values. That is why, what, what the greatest integer function is. If we plug in a decimal point for the X, we'll get an integer for the Y. So we can just put any, a number, the range won't be all real numbers. It'll be all integers. So our range will be the set of all integers. And just like that, we've obtained the domain in range for a graph. So all we're left to do is figure out which graph matches ours. We can eliminate answer choice C and D because they have the incorrect domain and range. And then we're left with answer choices A and B. And once we see our points, we would see that answer choice B matches better with the points that we have in our graph. They have a more specified X axis with more decimal points which we use and they have the correct domain and the correct range. So I hope that was helpful. And until next time.

Graph each function. Give the domain and range. ƒ(x)=[[2x]]

Key Concepts

Piecewise and Step Functions

Domain and Range of Functions

Graphing Floor Functions

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