Graph each function. Give the domain and range. See Example 3....

Question

Graph each function. Give the domain and range. See Example 3. g(x)=[[2x-1]]

Accepted Answer

Hello everybody. I'm night today. Today we're going to be looking at this question that states graph the greatest integer function given below and write the domain and range of the function. Now, the greatest integer function that was given is a of X is equal to the greatest integer function of four X minus one. So before we start with this question, we have to ask ourselves what is the greatest integer function? Well, let's write that we have that the greatest integer function, which is denoted by two brackets. We have the greatest in of function is open bracket, close bracket. So the greatest interior function is denoted by brackets. So we have the greatest in interior function of X is equal to N where N is less than, or equal to X is less than N plus one. Now, this rule will be a little bit more clear once we actually start putting some numbers into this. So that's exactly what we'll do. We'll start plugging in different values for X and obtaining a different value for the Y four hour graph. So we'll treat this like any other graph. And what I'd like to do when plot plotting points is making a T chart. So I'll have an X on the left column and Y values on the right column. And I'll test different Xes specifically negative 0.4, negative 0.3, negative 0.2, negative 0.1 0. 0.2 0. 0.40 point five and 0.6. So now is when we'll test and we'll actually put numbers into our greatest interior function. So we know how it works. I will specifically show you how to do negative zero point an X of the negative 0.4, the X at zero and the X at 0.3, the rest of the values I will plug them in as I previously calculated. But you can watch me solve for negative 0.40 and 0.3 so that you can solve the rest by yourself. So let's start with negative 0.4. We'll plug this into our function and we'll have that F of negative 0.4 is equal to open bracket. Denoting the greatest center of function four, multiplying negative 0.4 minus one enclosed bracket. Now, if we have four multiplying negative 0.4 minus one, when you plug that into our calculator, we'll get negative 2.6. And this will function as our X in our greatest interior function. So now we look at our function, we need to find an end that is less than, or equal to X, that is less than N plus one. So our ax is negative 2.6. So what could be our end here? And remember we have to look for the greatest integer. So we're dealing with integers. So the integer that would go here as our end would be negative three. So if we put negative three as our N, we'll have negative three is less than, or equal to negative 2.6, which was our X and negative three plus one is negative two. So we have that negative three is less than, or equal to negative 2.6 is less than negative two. So we have that our N is negative three. Therefore, our Y for an X of negative 0.4 would be negative three. So that is how you would use the greatest interior function at an X of negative 0.4. Now I will plug in the values up to the X of zero where I will show you once again how to do F of zero. So if we have an X of negative 0.3, we'll once again have a Y of negative three, add an X of negative 0.2, we'll have a Y of negative two, add an X of negative 0.1. We'll have a Y of negative two again. And now we have arrived at the X of zero. So once again, we'll have F of zero now and that'll be equal to open bracket or the greatest interior function of four multiplying zero minus one and close that bracket. So now we have four multiplying zero, which is zero and zero minus one is negative one. So we'll be solving for the greatest integer function of negative one. So now we have to look for the end that is less than, or equal to X, which is less than N plus one. Our X here is negative one. And remember we have to note that the very first symbol we have N is less than, or equal to X. So we can simply put negative one because negative one is less than, or equal to negative one. And N plus one here in this case would be negative one plus one which is zero. So we'll have that negative one is less than, or equal to negative one is less than zero. And this is accurate. So we have that our N in this case is negative one. So at an X of zero, we have a Y of negative one. Now once again, I'll plug in some values up to 0.3 and you can take some time to do the rest by yourself. So at an X of 0.1, we have a Y of negative one once again. And at an X of 0.2, we'll have a Y of negative one once again. And now, we've reached an X of 0.3. So we'll have F of 0.3. It's equal to open bracket of four, multiplied by 0.3 minus one close bracket. And that'll be equal to, if we plug, if we multiply four by 0.3 and then subtract the one, we'll be left with 0.2. So we'll take the greatest integer function of 0.2. So once again, looking at how we do that, we have to look for an N that is less than, or equal to 0.2, which is N less than N plus one. In this case, that N will be zero because zero is less than, or equal to 0.2, N zero plus one is one and 0.2 is less than one. So we have that our N, in this case is zero. So an X of 0.3, we have a Y of zero and now I'll plug in the rest of the values where you can take some time to try to solve them by yourself. At an X of 0.4, we have a Y of zero. And at an X of one of 0.5, we have a Y of one. So an X of 0.5, we have a Y of one and an X of 0.6, we have another Y of one. So now we can go ahead and start graphing we've gathered enough points to be able to draw a graph. So we'll draw a Y axis and an X axis and we'll start plugging in points. So I'm going to draw each dash on the X axis will denote a 0.1. So the first dash will be 0.1. You'll have 0.20 point 30.40 point five, 0.6, 0.7. And I'll extend the Y axis a bit to have 0.8 as well. Now, on the opposite side. So for, so the negative side of the X axis, I'll go up to negative 0.5. So I'll have negative 0.1, negative 0.2, negative 0.3, negative 0.4 and negative 0.5. For the Y axis, I'm going to go in increments of one. So the first uh dash I'll draw is just one. Then we'll have two, three and four. And I'll do the same thing on the negative side of the Y axis, I'll have negative one, negative two, I have three and negative four. So now I can start plugging in points. I have that at an X of negative 0.4, I have a Y of negative three. So I'll draw a point there. Now I also have that an X of negative 0.3, I also have a Y of negative three. So I draw a point there and I'll connect those two points with a straight line. Now we'll have a space between those two points and our next two points, we have that an X of negative 0.2, we have a Y of negative two and an X of negative 0.1. We also have a Y of negative two. And I connect those two points with a straight line as well. So between the first two points and these next two points, there is a space, there is not a continuation between those two points because those two points are connected by a horizontal line, the both points in each case. So then I can go to my next set which would be at an X of zero, which is Y of negative one at an X of 0.1, there's a Y of negative one as well. And at the next episode, 0.2, we also have a Y of negative one. So then I can draw a horizontal line connecting those three points once again having a space between this set of points and the previous. So then I can go M one to X of 0.3 has A Y of zero and X of 0.4 also has a Y of zero and those two points will be connected by a horizontal line as well. And then we have an X of 0.5 has A Y of one and an X of zero point. Six has a Y of one and I connect that with a horizontal line as well. So we can see the pattern, uh it almost looks like a line moving upwards. However, there is, there's spaces between each set of points. We have a horizontal line connecting each set of points. But the pattern is it's moving upwards. So that would be it for the graph. And now we can look at the domain and the range. So the domain deals with the possible X values in our graph. So let's look at that. So the possible X values here, we've plugged in anything, any X on the negative side of our X axis and on the positive side of our X axis, we can plug in any decimal point as we did. But we can also plug in any integer, we can plug in any possible point. So for our domain, we'll have the set of all real numbers because we are able to plug in decimal points, we're able to plug in negative values. And as long as it's a real number, we can plug it in. Now for the range we're dealing with possible Y values in our graph. So how this graph, how we move up and down in our graph. Now we have to notice if we look at our T chart for any X value that we put or that we plugged in, even if it was decimal points, we always obtain an integer. Which goes along with the greatest integer function. That is the point of the function. So we will any whatever point we plug in as our X, we will always, always, always obtain an integer as our Y. So we can't have once again the set of a real numbers because it isn't a real numbers, it's only the integers. So our range will be the set of all integers because those, those are the only possible whys in our function because of the greatest integer function. So just like that, we were able to graph our function as was figure out its domain and its range. So I hope that was helpful. And until next time.

Graph each function. Give the domain and range. See Example 3. g(x)=[[2x-1]]

Key Concepts

Piecewise and Step Functions

Domain and Range of Functions

Graphing Functions with Transformations

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