So for this problem, they give you a set of data to plot, and I'm gonna not gonna plot that. But we're gonna come up with the model. And so, uh, so we do have profit is a function of X. That is true. And X is the number of units sold, number of units solved. And they tell us that if the number of units sold is less than 100. So if X is less than or equal to 100 we know that it costs them. Uh, costs $60 to produce one of these units, and they sell it for $90. So they're gonna end up. If I look at their profit for this setting, we know that their profit will end up being the 90 times how much they bring in. So they're gonna bring in $90 for each item. However, it cost them 60 times. Uh, X is their cost. So this is how much it will. They'll bring in. So the revenue and then this is minus the cost. And then that's gonna end up being their profit. So we know that that simplifies Donald will simplify that in a minute. But what do we do if if we have an object that is more than 100 and if it costs more than 100 then they say they reduced the price. And so we know that, um, normally the price of an object is $90. That's how much they sell it for. But they're going to reduce that fit by 15 cents for every unit over 100. So we need to find out how money over 100. So if we take how many units and we already know the number is bigger than 100 if we're using this part of the piece wise function. And so let's say they were selling 101 objects, then we would need to subtract away 100 from that. And that would be the number of items that are over 100 and they're going to reduce the price by 15 cents for each one of those items over. So if they had to say this was 110 then 10 times they're going to reduce the price 15 cents. So if we put 110 in here 110 minus 100 would be 10, and we'd multiply it by the 15, and they'd reduce the price by a dollar 50. So this is what the selling prices but we need to sell. Multiply it by X, and that's going to end up giving. This is our revenue well, and I need small room. So let's erase this guy and we'll move him over a little bit because I leave myself enough room. Here we go, so we'll go X is greater than 100. So this is our revenue when you need to attract away the cost, will. The cost is always staying $60 for each item. So now what do we need to do? We need to clean these up, and we've We've satisfied actually what they asked us to do to write a function for this in terms of the number of units sold. But we can clean this up, So let's do that and subtracting these two, we would get 30 acts. That's what the profits going to be as long as you have something that is from zero to 100. Yeah, and then let's clean this up. We'll clean this up just over here, so that I don't have Thio mess with this down here. So we would be distributing this through so we would have 90 minus 0.15 x and then a negative times a negative is positive. And that 0.15 times 100 would move the death, the place I'd become 15 on then that needs to be multiplied by X and then we need to attract away 60 X. All right, so we can combine these two together and those two ed upto 105 and then we're going to distribute the X. We have 105 Oops, 105 x and then x times. This gives me minus 0.15 x squared and then we're subtracting away 60 x. So now we can combine these two together and then I'll write this final answer down here. So we have negative 20.15 x negative 0.15 x squared. And then again combining these two X values together, we're gonna get plus 45 X and that's going to be when X is bigger than 100. And so really, we have whole numbers for our domain. Yeah, because we can't have negative numbers and you can't sell portions of units. So really, our whole numbers would end up being our domain. And this is a function. And you could even check this function by going in and plugging in 1, 10, 1, 21 30 as the table shows to verify that this will will actually work. Now there may be a cap on how many, because realistically, they probably just can't make you know billions and billions of these things. So they're going to be restricted on how maney they can actually manufacture. So there's probably a cap on how largest would be. But you definitely do wanna have whole numbers, something where you're not in allowing decimals to come in. It's not just really numbers, but you will need a piece wise function.