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

Question

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

Accepted Answer

Hello everybody. How we doing? All right. Today today we're going to put again this question that states graph the greatest integer function given below and write the domain and range of the function where our function is F of X is equal to negative greatest integer function of five X. So the negative is outside and multiplying the greatest integer function. Now the answer to choices provided all show us different types of graphs with dash lines, meaning there's multiple X values with the same Y value horizontal dash lines that extend downwards or upwards along the graph. As choice A gives us a domain with the set of audio numbers and arrange with the set of all integers. And the la the dashed lines go downwards as a choice B has the domains with a set of all real numbers and arrange with the set of all integers and the horizontal lines. The pattern they go through is moving upwards in the graph. Choice C has a domain with the sale of all integers and arrange with the set of all real numbers with the dash lines following a pattern going downwards along the graph and answer choice D has a domain with the cell uh integers and arrange with the set of all real numbers with the horizontal dash lines following a pattern going upwards in the graph. Now, for this question where they specifically ask us, and we're specifically dealing with the greatest integer function. We first have to ask ourselves what is the greatest interior function? So let's recall that the greatest into your function can be written as open bracket, X close bracket. So the greatest inter function is represented by brackets is equal to N where N is less than, or equal to X is less than N plus one. So once I start testing this with different points, it'll make a bit more sense. And that's exactly what we'll do. Then we have to treat this like any other graph that we would, meaning we're gonna try different points on the X and see what Y values we get. And then kind of plot that on the graph. And to do that, I always like to do A T chart. So I have X values on the left column and Y values on the right column. And for this question, I'm going to test negative 0.7. These are all X values. So I'll test negative 0.7, negative 0.6, negative 0.5, negative 0.4, negative 0.3, negative 0.2, negative 0.10. And then all those same values again, but positive. So 0.10 point 20.30 point 40.50 point six and 0.7. So we are testing a lot of X values here and I'm going to do a couple of them. So you can see how this works with the greatest integer function, the formula that we have and the rest you can try and do them by yourself specifically, I'm going to test and walk you through negative 0.7, negative 0.4 and 0.3. So first, I'll test negative 0.7. So I'll have F of negative 0. and that will be equal to negative multiplying the greatest integer function of five, multiplying a negative 0.7. Since all we're doing is substituting the X with negative 0.7 and close the greatest in your function. And we'll have that this will be equal to negative well, and the greatest interior function of negative 3.5. So let's plug that into our formula for the greatest interior function we have that we'll have that blank is less than or equal to negative 3.5 is less than blank. So this is where we have to find the correct integer. Here, we have to find an integer that is less than or equal to negative 3.5. So since 3.5 isn't negative 2.5 isn't an integer. We're looking for one that's less than, and then when we add one to that integer, that we find it has to be greater than negative 3.5. So that integer in this case is going to be negative four. So negative four is less than or equal to negative 3. and negative four plus one is negative three, which is greater than negative 3.5. Therefore, our N here is negative four. And remember that we have a negative multiplying the end. So we'll have that negative one times N is four. So at an X of negative 0.7, we'll have a Y of positive four because we are multiplying it by one or by negative one. So that is how you use the greatest in interior function with negative 0.7. Now I'm going to plug in the values for negative 0.6 and negative 0.5 before showing you how to do a negative 0.4. So if you want to take some time and try to do negative 0.6 and negative 0.5 by yourself before looking at the values. So when you have an X of negative 0.6, you would have a Y of three. And when you have an X of negative 0.5, you will have a Y of three again. And now we've reached negative 0.4. And once again, I'll show you how to do that one. So we'll have F of negative 0.4 is equal to negative multi outside of the greatest interior function of five, multiplying negative 0.4 and close the greatest into your function that will be equal to negative multiplying the greatest in interior function of five multiplied by negative 0.4 is negative two. So now we apply this to the formula that we had. So we'll have blank is less than, or equal to negative two. It's less than blank. So what integer is less than or equal to negative two? Well, that is negative two because negative two equals negative two. So negative two plus one is negative one. And now we have that negative two is less than or equal to negative two is less than negative one. So R N is equal to negative two multiplied by negative one. Remember we have a negative multiplying our greatest interior function. So our N is negative two multiplied by negative one, which is positive two. So add an X of negative 0.4, we have a Y of positive two and now I'll plug in the different values up to 0.3. So once again, if you want to take some time to try that by yourself before we try again with 0.3 together, so that an X of negative 0.3 will have a Y of two. Once again, add an X of ne negative 0.2, we'll have A Y of one, add an X of negative 0.1. We'll have a Y of one. Add an X of zero. We'll have a Y of zero, add an X sub 0.1 will have A Y of zero at an X of 0.2. We have a Y of negative one. And now we'll try F 0.3 together. So at an X of 0.3, we'll have F of 0.3 is equal to negative multiplying the greatest interior function of five, multiplying 0.3. And that'll be equal to negative multiplying the greatest interior function of five multiplied by 0.3 is going to be 1.5. So we'll have that blank is less than, or equal to 1. is less than blank. So now we have to look for an integer again. So what integer is less than 1.5? And when we add one to that integer, it'll be greater than 1.5. Well, the integer is one, one is less than 1.5 and one plus one is two, which is greater than 1.5. So we have that N is equal to one multiplied by negative one. Remember we have a negative multiplying our greatest interior function, which means that we'll have negative one as our answer. So at an X of 0.3 will have a Y of negative one. And now I'll plug in the remaining values. But once again, you can use the technique we've been using to try them by yourself. Set an X of 0.4 will have a Y of negative two at an X of 0.5. We'll have a Y of negative two, add an X 0.6. We have a Y of negative three and an X of 0.7 will also have a Y of negative three. So now I'll scroll down a bit so I can write down all of the points we've obtained and those points are going to be negative 0.7 comma four negative 0.6, comma three negative 0.5 comma three negative 0.4 comma two negative 0.3 comma two negative 0.2, comma one negative 0.1 comma one zero comma zero 0.1, comma zero 0.2, comma negative one 0. comma negative one 0.4 comma negative two 0.5, comma negative two 0.6 comma negative three and 0.7 comma negative three. So now let's talk about the domain and range. So the domain is all the possible X values of our graph. And as we can see, we plugged in negative decimal points and negative and positive decimal points as well as zero. And we can continue to do this infinitely to the left and to the right, there is no X here that we would not be able to plug in. So our domain which deals with possible X values would be the set of audio numbers because we are able to plug in any real number into our X. And it would give us a proper answer. Now, for the range, it's a bit different because the range deals with different Y values. And as we said, there's multiple XS with the same Y value. Now, since we're dealing with the greatest interior function, we know that we're dealing with integers. So we were able to plug in decimal points for the XS, but we always have an integer as our answer for the Y. That means that our range will be the set of all into yours. We can't put all real numbers because that's not true. We never will obtain a decimal point as our answer. So we can look at are the answer choices provided and we can eliminate some answer choices. For example, a choice C has the incorrect domain and range. So we can eliminate answer choice C and we can do the same for answer choice D which has the incorrect domain and range. Now, we're left with looking at answer choice A and B and that we see that they each have a different pattern. Answer choice A has a pattern with the dash lines moving downwards while answer choice B has a pattern with the dash lines moving upwards. Well, let's look at our values. We see that as we move, we look at our T chart now and we see that as we move from negative axes to positive axes, our Y goes from positive wise to negative wise. Therefore, our Ys are moving downward. Meaning that answer choice A is the best answer for our question. It's going to represent the graph of the greatest integer function of our F of X function with a domain of the set of all real numbers and a range of the set of all integers. So I hope that was helpful. And until next time.

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

Key Concepts

Graphing Step Functions

Domain of a Function

Range of a Function

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