before we dive into new formulations of functions it helps to take a step back though and ask a more general question about what is a function in the more general sense and what are the boundaries of what defines a function what are the things that we can play with and what are the things that we can't going back to the original definition you probably saw functions is that a function is a rule or process that assigns to each input a corresponding output and there's a little subtext here that's usually implied by the context but there's a maximum of one corresponding output in other words we can only get one output not two or three or four or five and some inputs might not have an output at all like square root of negative one does not doesn't have a real output with that idea of an input being translated into a single output then what we can look at is how these kinds of functions might be expressed or formulated so let's take a look at the example on the next page fairly simple we're going to have a wheel of radius one and when we draw that out just to be clear here the radius is always one and we have a green dot g and what we're going to be doing is turning the wheel counterclockwise by an angle of theta so we're going to be going in this direction here and the output here's our input here our output is going to be the height so there's a lot of things we could use as an output we are specifically going to use the height of the point so for example if we were to rotate to this position here this would define a range or an angle here of theta and the h that we're interested in would be this vertical height drop down here h of theta now if we take a look at that construction then there are various ways we can try to examine how theta and its output are related by this function by this verbally described function one thing we might try at first is simply to sketch a graph of this relationship so we draw ourselves some axes here and label them as theta as the input and h of theta as the output we know that we can sketch a few sample lines here and when we know that theta is zero the height is going to be zero because we're not elevated above the line there then we can erase that and imagine a little later on let's take pi over 4 which is 45 degrees so then we have a theta value and we're going to get a value back for that well we know the radius is one so it's going to be something a little less than one so let's put one in our chart here and at pi over four we're gonna have something a little less than one actually that's making me think another way to tie these two together might be a table where we have our theta and we have our h of theta and we put in some sample values here we're using radians because we're in a calculus class as opposed to degrees keep that in mind that'll be our common thread as we go through this uh three pi over four we can compute some height values for these and if you remember your trig off the top of your head you'll recognize that the side length here is in fact the sine of theta the height and so we do these calculations and we can get sine of theta the sine of theta when theta 0 is zero sine of pi over 4 would give us 1 over root 2. when we get to pi over 2 erase these if we go right to 90 degrees pi over 2 then we're going to have of course a height of exactly 1 and that's going to be in here and when we go even further we're going to get back to 1 over root 2 and so on in fact we're actually going to go negative afterwards with that construction we can actually build our graph out and we know in fact when we reach pi which is the same as a half circle in our radian measure we're going to be able to return to a height of zero when we move our green dot all the way around and then we are going to go and do the same thing but in the negative heights like so then we simply extend our graph to match that and we would end up back here at 2 pi and if we wanted to there was no restriction given so we can imagine this repeating itself forever as we simply watch the green dot traverse around that circle with that simple exploration we've actually highlighted most of the ways that engineers and mathematicians use functions if we take a look at what we have here we had our verbal description we built a graph either both as a sketch or as a more detailed plot using sample points and we actually defined through trigonometry a formula for that relationship if we know the angle we can compute the height of the point on the circle as it goes around the trajectory and that height is defined simply by the sign of that angle in radians and so we get this collection of tools that we can use when we're trying to describe functions the idea is to emphasize that a formula which is usually what we think of as a function or identified to the function in a mathematics class is not the only way to describe functions very frequently we are going to have tables of data coming from some kind of sensor information engineering and that is going to be the definition of our function in a way that may not even be available as a formula graphs can also give us some insight into the structure of a table of data or a formula help us to understand what we're looking at in a different way and of course the verbal descriptions are usually where we start in fact those are the definition of the problem and then we go and dive a bit deeper and get more quantitative information through the other representations but the key thing is don't limit yourself to thinking of functions as simply formulas there are lots of alternatives out there so that idea captured that we shouldn't limit ourselves simply to formulas we still want to keep this general idea of a model of a function where we have an input and an output that is going to be consistent across a variety of different ways we can develop formulas just a reminder about terminology we would call the output here the dependent variable and the input variable over here is what we consider the independent one we don't have any control over time time marches on but and so it's independent of our choices the dependent variable depends back on time we won't use that phrasing that frequently in this class we'll typically talk more about inputs and outputs but the phrasing will occur from time to time it's just important to have that in mind especially when you do experimental courses like mod 2 then you'll definitely be using this kind of dependent and independent terminology now let's go back to our canon of knowledge that we already have and we're going to take a look at functions that you're familiar with and note specifically that by and large these functions would have single numbers for inputs and outputs for example let's do y equals x squared we would have a single x value and that would give us as an input rather and that's going to give us a single y output for each x there's just one dimension of input and one dimension of output and of course that same thing applies if we'd written it as f of x equals x squared and function notation same idea think of the other functions we've seen exponentials like e to the t x equals e to the t again one input one output y equals cos of x or cos of 10x let's change it up and other functions logarithms g of x equals lon of x in each of these cases we have one single value we put a single number in for x we get a single number y back and we can then graph that by plugging in a range of x values to get a range of y values back and that would give us both tables and graphs if we wanted them in the upcoming videos we're going to see how we might expand this idea of a function so that we have more than one input maybe have an x and a y come in and give us a z back or alternatively what if we had time coming in and we get not just an x value back but maybe a separate y value what would those kinds of formulas look like and how would that play into the calculus rules that we've already used that'll be what we're exploring for the rest of this topic