 In previous examples, we've seen how we can use modeling of functions to predict the future. It can go the other way around as well. If we have an accurate model, we can try to predict what happens in the past, whether it's like a historical thing, anthropological thing, maybe like a crime forensic type thing, we can predict what happened based upon our mathematical models. So predictions are a useful tool, a useful application of these story problems and functions here. But sometimes it's not about prediction. Sometimes we have choices in front of us. We have one choice versus another. And we can use function modeling to actually decide what is the better choice. So case in point, take Jamal here, who's choosing between two truck rental company. He's gotta move from his old apartment to a new apartment. And so he's got some stuff. He's gotta rent a truck to move his things from his one apartment to another. And so his local town has sort of two competing renters, rent truck rental companies he could use. The first one called Keep On Truckin Incorporated. And it charges the use of the truck. It's not on a timeframe, it's mostly just on usage, right? And so they would charge Jamal, first of all, a $20 fee. So however much he uses that day, it doesn't matter. He'll be charged a $20 rental fee. And then he'll also be charged $0.59 per mile. And that's not incorporating the gas that he has to put in there. This is just the usage. He'll be charged by mileage and he'll be charged just for signing out the rental agreement. The competitors to Keep On Truckin is Move It Your Way. They also charge an upfront fee of $16, but they charge 63 cents per mile. So we can already see right there, a sort of a competition there. Keep On Truckin has a higher upfront fee, but it has a slower mileage. Move It Your Way is the opposite. It has a smaller upfront fee and it's gonna be 63 cents per mile. It's still a bit more expensive. So there are some advantages and cons to the two different companies. There's not an obvious choice that's better for Jamal. We need some more information. So what we know is that Jamal has to, he has to drive 50 miles with the truck. So he's calculated what would be the distance to drive the truck from the rental company to the old apartment, drive it to the new apartment and then drive it back to the rental company. That turns out to be 50 miles. Now that might seem a little bit weird because next to each other, well, just for the sake of example, we'll say that. So Jamal would have to travel 50 miles in the truck. And so who's the better option if he has to drive 50 miles? Well, if you wanted to do the first, the first moving company here, the first rental here, keep on trucking right here. What's the cost function associated to keep on trucking? Well, from the information we're given here, if we say that F of X is the cost function associated to renting a truck from keep on trucking, well then they're gonna charge you 59 cents per mile. Let's let X equal the number of miles that Jamal has to drive here. 59 cents per mile and there's a flat fee of 20 miles. This would tell us that F of X is gonna equal 59 cents times X plus 20. So notice here that X is gonna be miles. The rate 0.59 here is gonna be cost per mile. So when you take miles times cost per mile, you get cost, add that to cost. This is gonna be the cost function for keep on trucking. We're just gonna call it keep here. And then conversely, if we wanna do move it your way, their cost function would look like the following. You're being charged 63 cents per mile plus 16 miles. And so move it your way has this cost function we know that Jamal has to drive 50 miles like so. So he's gonna calculate these costs here. If he looks at F of 50, this would be 0.59 times 50 plus 20 here. And you could do all, I mean all this arithmetic, we could pen and paper right here, but you know, don't be a hero, you can use your calculator to help you out here. 59 cents per mile times 50 miles, that would be $29.50. Add that to the flat fee of $20. He would expect to pay $49.50 if he went with keep on trucking. On the other hand, if he does G of 50, this is gonna be 63 cents times 50 plus $16. 63 cents per mile times also be $31.50. Add the flat fee of $16.00. That turns out to be $47.50. So assuming the two companies are pretty equal except for their cost functions, right? You know, assuming like the quality of the equipment, the reputation, the brands or all this, it looks like Jamal would save a little bit of money, not a lot, but he would save $2.00 by working with Move It Your Way instead. And so it would seem like that's the better choice for him, he should be using the second company for a way, it's pretty tight though, right? It wouldn't, I mean, $2.00 is not a huge differential, but that does seem to be the winner in this situation. All things created equal, I would choose the cheaper option there, which would be Move It Your Way. But we can see it's pretty close. And so another thing to ask here is at what point does the first rental company keep on trucking actually offer a, when does the first company offer a better deal? So basically what we're asking is the following, right? We want to solve, we need to solve the inequality f of x is less than g of x, right? So we could do it the other way around as well, right? When is f of x greater than g of x, it doesn't make much of a difference, but we have to compare the inequalities going on right here. And so when you have an inequality, 0.59x plus 20 is less than 0.63x plus 16 here, right? We want to solve this inequality. Like I've mentioned before, when you solve an inequality, you basically want to start off by solving the equation, right? We're basically trying to figure out geometrically, where do these two lines intersect each other? And so if we think about the graphs of these functions right here, one of the functions is going to look like this. It has a b, but a smaller slope. And then the other one has a smaller initial fee, but it has a higher slope. At some point, these things are going to intersect themselves, where one is going to be bigger than the other, right? And so that's this point of intersection is going to be interest to us. That comes down to solving the equation. So we're going to take 0.63x minus 0.59x, and this is going to equal 20 minus 16. If you start solving this linear equation, the right-hand side's pretty easy. That should be a four. 69 cents, 63 cents, excuse me, so it's going to be four cents times x. So divide by both sides, you're going to get that x here, x is equal to 100. So if you drive more than 100 miles, so 100 miles is the point match right there. At 100 miles, the two companies will cost the exact same in terms of renting. And so that's going to be the tipping point. We've seen already, we already did a test point here. We saw already that when you're less than 100 miles, move it your way is the superior one. I think I have my colors backwards right now. So the cheap, no, this is right. So move it your way was green and it's cheaper as long as you're less than 100. But when you get past 100 miles, keep on trucking is going to be the cheaper option here. And so therefore, if Javel was moving less than 100 miles, he should use move it your way. If he's moving more than 100 miles, he should do keep on fuel. And that's how he can make his decision is if he turned out like, oh yeah, I only did one way, I'm going to have to do multiple trips. Like maybe he has to drive, he has a lot of stuff. Maybe he has to, and it's a small truck, right? You might have to drive between the departments two or three times. And that actually might add up the mileage. And so you might find out, oh, actually I'm going to have to drive 120 miles to do make this move. In which case then keep on trucking could be the better option. And we see that the function modeling is done effectively. It helps us answer these questions by having specific functions. It helps us understand what's going on here. Linear functions are simple enough that we might not actually need to come up with algebraic formulas, algebraic equations, algebraic inequalities to answer these questions. But be aware, the goal is to be good at algebra so that when we do get to situations, which are not easy, then we can still use the techniques we've practiced with linear functions in that setting as well.