Let's do some example problems dealing with functions and their domains and ranges. Just as a review, a function is just an operator-- let's say this function is f; that tends to be the most typical letter for functions-- that operates on some input, in this case, the input is x, and it produces some output y. Or you could view it as you take some input x, put it into your function box f, and it's going to operate on it and produce some y. And the set of values that x can take on is the domain. The set of values that f can legitimately operate on, that's the domain. And the set of values y that f can produce, that is the range. Now, with that in mind, let's figure out, one, the function definitions for each of these problems here, these example problems, and then figure out the domains and the ranges. So here we have Dustin charges $10.00 per hour-- so let me write that down--$10.00 per hour for mowing lawns. So how much does he charge as a function of hours? So let's say Dustin charges, so the function, I'll call it d for Dustin. Dustin charges as a function of hours. It depends on hours. The input into our box is going to be hours that he worked. It's going to be equal to the number of hours times 10. It's going to be 10 times-- I won't write the times there. Maybe I can just write 10h. And, of course, he can't work negative hours, so we could write h is going to be greater than or equal to 0. That's our function definition. If you say, hey, how much is he going to charge for working half an hour? You put 1/2 in here, which essentially substitutes this h with a 1/2. You do 10 times 1/2. Let me write that down on the side here. So Dustin's going to charge for 1/2 hour. He's going to charge 10-- wherever you see the h, replace it with 1/2-- times 1/2, which is equal to $5.00. That's our function definition. I was just showing you an example of applying it. Now, let's figure out the domain and the range. So I almost explicitly say it here, you can't work negative hours. This number, this function, is not defined if h is less than 0. So we could say non-negative. And we could say real numbers, which it could be everything including pi and e and all of that. But you can't legitimately charge someone 10 times pi dollars. That's not a legitimate amount that one can charge someone in the real world, because at the end of the day, you have to round to the penny. You can't actually charge them that. So any number that he charges is going to be a rational number. It can be expressed as a fraction. So we can kind of take out numbers like e and pi. So if we want to be really cute about it, we would say non-negative. rational numbers. And these are just numbers that can be expressed as a fraction, which are most numbers, just not these numbers that just keep not repeating and all of that. So non-negative rational numbers is what he can input in here. And then the range, which is the valid-- once again, whatever number you put in for h, remember, you can only put in non-negative values for h, non-negative rational values for h, so whatever numbers you put in for h, you're going to get positive values for how much he charges for 10h, for the value of the function. So once again it's going to be non-negative, and if you put a rational number here and it's being expressed as a fraction and you multiply it by 10, it's still going to be a rational number. And if you want to be really cute, you could say any number that could be expressed as a dollar sign. But anyway, I don't want to get too cute on this problem. I think you get the idea. Let's do more problems. All right. Here we have Maria doing some tutoring. Maria charges$25.00 per hour for tutoring math with a minimum charge of $15.00. So Maria charges as a function of hours. So she's going to charge$15.00 if you don't get enough hours in. So at $25.00 per hour, in order to make$15.00, you've got to work 15/25, so it's 3/5 of an hour. So if her hours are less than 3/5 and greater than or equal to 0, she's going to charge $15.00. Because if she only worked 1/5 of an hour, the bill would have been$5.00, but she says she has a minimum charge of $15.00. So she's going to charge$15.00 up until 3/5 of an hour, or 36 minutes. And then after that, for h greater than 3/5, she's going to charge $25.00 an hour. She's going to charge 25 times h. So that's her function definition right there. Now, what is the domain, the domain for Maria's billing function? Well, once again, this is only defined for non-negative numbers. And by the same logic we used here, we got a little cute. We said, oh, she really can't charge e hours or pi hours. That's not realistic for her to measure. Everything's going to be expressible as a fraction. For example, even 5, if you say 5 hours, that's still a fraction. That's 5 over 1. So let's say non-negative rational numbers. And what's the range of her charging function or her billing function? Well, it has to be at least$15.00. No matter what, she's going to charge $15.00, so it's going to be$15.00 or more. So it's going to be rational numbers greater than or equal to $15.00. There's no situation in which she will charge$14.00. There's no situation in which she would charge negative \$1.00. Everything's going to be greater than or equal to 15. So now we have these more abstract function definitions, so now can stick really any number in it. Well, we're not going to deal with complexes, any real number. But let's see if they limit it a little bit. So we have f of x. Let me rewrite it. f of x is equal to 15x minus 12. So what's the domain here? What's our domain? Well, I can stick any real number x in here and I'm just going to multiply it by 15 and then subtract 12. I could put pi there. I could put the square of 2 there. I could put e there. I could do all sorts of crazy things. So I'm going to say all real numbers. Any real number I put in for x, I'm going to get a legitimate output. And then what is my range? Well, once again, this can take on any value out there. I can get to any negative value if I make x negative enough. I'm going to subtract 12 from it, but I could get to any negative value. I could get to zero. I could get to any positive value. So this is also going to be all real numbers. Now we have this function. It says f of x is equal to-- they forgot the equal sign-- 2x squared plus 5. So what is the domain? What are all of the valid values for x that I could stick here? Well, I could stick anything here. I could put e there. 2 times e squared plus 5. I could put a square root of 2 there. I could put a negative number there. So I could really take on any real number. All real numbers, positive or negative or otherwise, or even non-rational. And then what is the range here? And this is interesting. Because no matter what number I put here, this value right here is going to be greater than zero. This is going to be, right here, non-negative. Even if you put a really negative number here, you're going to square it, so it's going to become a really positive number when you square it. So this expression right here is going to be non-negative. So if you take a non-negative number and you add it to 5, you're always going to get something greater than or equal to 5. Worst case, this is a zero, and then you get a 5. But if this is either slightly less than zero or slightly more than zero, you're going to get values greater than 5. So the range is all real numbers greater than or equal to 5. Your f of x could never take on the value zero. There's no way to get zero. Problem 5. They have f of x is equal to 1 over x. So, once again, our domain. Well, we could put any value in for x. Any real number will work here. We're just going to take the inverse of it. So this is going to be all reals, all real numbers. And then what is the range? This is interesting as well. Because no matter what I put here, I can put anything here. Oh, actually, let me be very, very careful. This is not defined for x is equal to 0. All real that are not equal to zero. If I put zero here, I'm going to get 1/0. It will be undefined. Let me write that down. I almost made a mistake. f of 0 is undefined. And that literally means we don't know what to do with the zero. We have not defined a way to operate on a zero. So f of 0 is undefined. So the domain is all reals except for zero. And now what's the range? Well, once again, we can take on any value here except zero. We could try to approach zero. If x got really, really, really, really, really large, then this will approach zero. If x got really, really negative, it'll approach zero. But in no circumstance will we actually ever get to zero. So once again, all reals that are not equal to zero. All real numbers except for zero. All right. Problem 6. What is the range of the function y is equal to x squared minus 5 when the domain is-- so they're defining the domain. They're restricting it. They're saying the domain is just the numbers minus 2, minus 1, 0, 1, and 2. So this function definition really should say f of x is equal to x squared minus 5 for x is equal to-- we could write any member for x is a member of the set minus 2, minus 1, 0, 1, 2. It's not defined for x being anything else to that. You don't know what to do with it. You can only apply it if x is one of these. So what's the range? Well, the range is all of the functions, all the values that f of x can take on. So the range is going to be the set of values that we get when we put in all of these different x's. So let's try this first one: negative 2. Negative 2 squared minus 5. That's 4 minus 5. That's negative 1. Let's do negative 1. Negative 1 squared minus 5. That is 1 minus 5. That is negative 4. You have 0. 0 squared. So f of 0, you have 0 squared minus 5, that's negative 5. And you have 1. 1 squared minus 5, that's negative 4, so it's already in our range. Then you have 2. 2 squared minus 5, that's negative 1, already in our range. So that's all of the values that this function can take on, given that we've restricted it to only these inputs.