Howdy folks. Welcome to this video about functions. Um, in calculus, we are often thinking about how two quantities are changing in relationship to each other. And functions are all about relationships between two different sets of values. Usually in a math class and a calculus class, both of those sets of values would be numbers. But that doesn't have to be the case. A function is really any relationship between any two sets of possible values but with the condition that every input value corresponds to exactly one output value. So you might think of a function as being a rule that tells us exactly what to do with each possible input. So a function is a rule or is a relationship where there's no ambiguity. We ask a question of our function and we get one answer back. We call the different possible inputs the domain of the function. So you can see here we have four different possible inputs and we call the possible outputs the co-domain. So to build a function we would take and say okay maybe our function says this input should correspond to this output and this other input can correspond to this other output. So when we input one thing we get out one thing. It's okay to have a situation like this where we ask a different question. We put a different input in, but we get the same answer as asking another question. Um, it's like we could ask what's 5 + 5 and we could ask what's 7 + 3 and the answers would both be 10 and that would be okay. What we don't want to have happen is if we have two arrows coming out of the same input. So like if we added a relationship between this input to a different output than we assigned it before. That would be like asking the same question and getting a different answer each time. So, we don't want that to happen. Some functions. So the functions we deal with in calculus are going to be functions where the domain and the co-domain are both sets of numbers. We can write the domain or the co-domain usually using interval notation if you like using interval notation. Um we can also use inequalities if you like using inequalities or we can just describe the domain and co-domain using words. We can say things like all the real numbers or all the real numbers except two. Um but a function is just a relationship between two sets of values that satisfies this condition where there's not any ambiguity with what to do with our input values. So we could make a function with something weird. We could make a function um out of the birthday month rule. So the birthday month rule says the domain the possible inputs are different people. So we're going to input a person and the outputs the co-domain will be the different months of the year. And if we input a person we output their specific birthday month. So this is a function because each person has exactly one birthday. So, we ask a question, we're going to get one answer. Here's an example with numbers. The depth rule. If we input a time and output the depth of water in a particular rectangular fish tank. So, our domain would be different possible times. Our co-domain would be the different possible depths for this specific fish tank because we're talking about one specific fish tank in particular. Using one fish tank makes this a function because at any specific time the water is only going to have one depth. We can't have the water be at two levels in a rectangular fish tank in the same time. The last example on this list is what we might think of as a function in mathematics and it's the sign rule. If we input an angle, we could output the y value corresponding to that angle on a unit circle. If we go around the unit circle uh counterclockwise that angle. So if we have a unit circle like this, we would start at 0 degrees and we could go around and we would report the ycoordinate would be the sign value. And the sign function is a function because if we have a triangle or if we have a unit circle, if we take an angle, it's going to have exactly one sign value. In contrast, here are some non-examples of functions. So again, we have an example where we work with something other than numbers, which is the pet rule. If we input a person, we could output the kind of pet they have. But that's not a function because a person could have more than one kind of pet. You're allowed to have a dog and a cat or a bird and a fish. One with numbers would be the elevation rule. if we input an elevation and output the time at which a person on a hike reaches that elevation. So maybe our rule is we're asking the question, hey, when were you at 1200 ft today? But that wouldn't be a function because a person could have an elevation at more than one time. Especially if you're on a hike, if you go up and then back down, you're going to have the same elevation at least twice. One example that we have to be careful about when we're doing algebra is the square root rule. So this is an again a non-example of a function. If we input a number and output its square root that sounds like a perfectly reasonable mathematical thing to do that should give us a definitive answer. But this isn't a function because a number has more than one square root. It has a positive square root and a negative square root. Um, the way we get around this is when we say the square root function, what we mean is the positive square root. Um that is all for this video but please do take a few minutes just to reflect and come up with some examples and non-examples of functions that fit um fit your circumstance that make sense to you.