 This video will talk about Peacewise Function Applications. So a local gas company charges 75 cents per therm of natural gas up to 25 therms. Once the 25 therms are exceeded, the charge doubles to $1.50 per therm due to limited supply and great demand. Write these charges for natural gas consumption in the form of a Peacewise Defined Function, C of T, where C of T is the charge for T therms. Let's see what we have. The first part says that we have 75 cents up to 25, and it says once 25 is exceeded, then it changes. So that means it's going to include the 25. So we have 0.75 times x, that would be the 75 cents per therm, but the domain is going to be from zero, including zero up to and including 25, and then it says once it is exceeded that, the charge doubles to $1.50 per therm to the limit. Okay, so now for the second piece, we know it's going to be $1.50 per therm, so it's 1.50x, but we have to consider that we have all this, we started, so we have to take that out. So if we take 0.75 times the 25, we're going to end up with 1875. So the most you're going to pay for 25 therms is 1875, so $1.50 times x minus to 1875 will be in the domain of x is greater than 25. Okay, so they want us to graph this. I'm just going to do a real rough sketch here. So I've got my C of T and I've got my T, and I'm going to go to my calculator. Okay, so I have these two equations in here, and I know that at 25, they actually should have the same value. Let's do this first part then, from 0.75 times 0 would be 0, be 0, and if I had up to 25, put that here, I'm going to have 18.75, and that goes here, and all of that is included, lying between the two. Now if I go back to my calculator then, I can see that these are going to be increasing for me, but how do they increase compared to the other one? So I look at my graph, this was Y1, this was the first part of my graph right here. So this part right here, we drew, and now it's going to be steeper as we go from that point on. That's really all I wanted to know. It's going to be an open circle, but it's been closed in from the first part, and then I've got to have a steeper looking graph going on forever. And then the last part says, find out when it's 45 therms, how much it's going to cost. It's going to be in the 1.50 times X, so X is 45, and then minus my 18.75, and if I use my calculator again real quick just to do the calculation for me, I can do second window, 45, and then second graph, and it's going to tell me right away that it is going to be the second piece, so it's 48.75, $48.75 for 45 therms. Alright, let's try this one. Missions price at Welling's Wet World are as follows. Infants under two years are free, then admission is charged according to age. So let's see, it asks us to write the piecewise function, so let's just write that as we go. So this is admission, so we'll call it E of T, T being age, and we have infants would be zero to two years, be free, but then the next one says children two and older, so this is up to but not including, so it's $0 for anything between zero and two but not including two. Children two and older but less than 13 pay $2, so then we've got a two, and that's from two including, up to 13. Teenage is 13 and older, but less than 20 pay $5, so we got a five here, and that would be including 13, up to 20, not including, and adults 20 and older but less than 65, it's going to be $7, so that's our 20 inclusive, and then less than 65, and then senior citizens 65 and older get the teenage rate, which the teenage rate over here was $5, so we go back to a five here for including 65 and greater. We've now written this and we've gotten all the domains for each piece going on then, so sketch the graph and find the cost of admission for a family of nine, which includes grandparents who are 70, they're going to be $5, two adults, and that was $7, and three teenagers, which are $5, and two children, which is $2 and one infant, which is $0, remembering what we had here, let's draw our little graph, this is going to be age, and this is going to be our output of age, so from zero to two, but not including two, it's going to be the x-axis, then we go from children who are two up to 13 but not including 13, so this is two, this is 13, it's going to be $2, now includes two but goes to 13 and doesn't include that, and then from children we went to teenagers who are five, three, four, five, and that's 13 up to 20 but not including 20, then we have $7 for adults and that was, seven was from 20 up to 65 but not including 65, and then we had senior citizens who were 65 and they started that back down at teenagers, which is five, so we come back here up here at five, close the circle, and then have an arrow, and now to figure out what we have to evaluate, we have two times five, plus we have two adults which is two times seven, plus three teenagers which is three times five, plus two children which is two times two, and then the one infant is zero, $43 for that family to get in.