Welcome to calculus. I'm Professor Ghrist. We're about to begin lecture one on functions. Welcome to chapter one. Every story begins with a cast of characters. In this story of calculus, our characters are functions. And the plot revolves around how these functions interact with each other and what characterizes them from the local workings of their inner behaviors, to the global sweep of their asymptotics. This lesson, and this course, begin with functions. Calculus is all about functions, and indeed, that is where this course begins. Now you're probably used to thinking about functions in terms of their graphs. The pair x with f(x) for a function f. One can also take a more mechanistic view, where x is considered as an input to f, and f(x) is considered as the output. Indeed, visualizing this as a machine is extremely helpful, where one feeds x into f and receives, as an output, f(x). With this in mind, certain terminology about functions becomes natural. For example, the domain of a function, f, consists of all possible inputs, all the things you might put in to the function. The range of a function, f, consists of all possible outputs, things you might receive from f. Now this is single variable calculus, and because of that, our domains and our ranges are going to be relatively simple. They are going to consist of either the real number line, or certain subintervals thereof. Certain operations on functions become critical. Perhaps the most important is that of composition. F composed with g. This is the function that takes as its input x, and returns its output, denoted f composed with g of x, and that is g(x) fed into the function, f. We say f(g(x)), and one can visualize this as chaining together the functions f and g in the proper order. g comes first, and then f. For example, if one considers the function, square root of one minus x squared, how can one decompose this into the composition of f and g? Well f, in this case, would be the function that is on the outside or that is done last. This is the square root function. g, that which comes first, is what is on the inside of the square root function. Namely g(x) equals 1- x squared. One additional important operation on functions is that of inverse. The inverse of a function, f, is denoted f with a superscript, -1. Don't let that fool you. That does not mean you take the reciprocal of f(x). Indeed, f inverse of x is defined as that function which takes as its input, x and returns as an output f inverse of x such that if one composes with f, one gets back the original value of x for all such x. And now, one could also run this in reverse. If you do f first, and then f inverse, one again obtains x. That means that, intuitively, f inverse is the machine that undoes whatever f does. Let's look at a specific example where we take as f, the function x cubed. What would the inverse of this be? Well, it has to be a function that undoes whatever x cubed does. This is, as you may have guessed, x to the one-third power or the cube-root of x. Now, there are several ways to see why this is right. One is that in taking the inverse, we are reversing the role of the input and the output or, geometrically, we are flipping the graph of this function along the diagonal line where the input and the output are equal. Of course, we can also think in terms of what the inverse has to satisfy. If we do x cubed and then take its cubed root, we get x. Or the other way around. If we take the cube root of x and cube it, we get x. Certain classes of functions wind up being extremely important throughout calculus. Perhaps the simplest such class is that of the polynomials. That is, functions of the form a constant plus a constant times x plus a constant times x squared, all the way up to some finite degree. Constant times x to the n. That largest power of x is called the degree. There is a summation notation that makes writing out polynomials very simple. We use the Greek capital, sigma. And write a polynomial as the sum, sigma, as k goes from 0 to n of c sub k times x to the k. Here, the cks are coefficients, or constants. Another class of commonly observed functions are the rational functions. These are functions in the form P(x) over Q(x), where P and Q are polynomials. Simple examples like 3x- 1 over x squared + x- 6 are very common throughout mathematics and its applications. Rational functions are very nice to work with. You do, however, have to be careful of what happens in the denominator when you try to plug in a value of x that evaluates the denominator to 0. Your function is not necessarily well defined at such a point. Other powers besides integer powers are important and prevalent. I'm guessing that you all know what x to the zero is. That is, of course, equal to one. What's x to the minus one-half? Recall that fractional powers connote roots, and negative powers mean that you take the reciprocal. So that x to the negative one-half is 1 over the square root of x. Now what is the x to the 22-sevenths? Well, break this up into pieces. First we take x to the 22nd power. Then we take the seventh root of x. Lastly, what is x to the pi? Well, we're not going to answer that quite yet, but you may have a guess. And especially considering the fact that there are rational numbers that are very, very close to the irrational number pi. Trigonometric functions are extremely common and important. You should already know a bit about sine and cosine. Let's review perhaps the most important, the trigonometric identities. That is, cosine squared + sine squared = 1. There are several ways to interpret this. You've seen the interpretation involving a right triangle with hypotenuse equal to 1 and with angle set to theta. Then, in this case, the sine of theta is the length of the opposite edge to theta, and the cosine of theta is the length of the adjacent edge. If we think of those as x and y coordinates, respectively, then we see that there's a relationship between this trigonometric identity and the equation for the circle. x squared plus y squared equals one. And indeed, if we think of what happens when we move a point along a unit circle, rotating it by an angle theta from the positive x axis, then the x and y coordinates of that point on the unit circle are precisely, cosine and the sine, respectively. Other trigonometric functions are common and important. I'm sure you recall the tangent is the ratio of the sine to the cosine function, and its reciprocal, the cotangent function. One can also take reciprocals of cosine, obtaining the secant function, and of sine, obtaining the cosecant function, respectively. Now all of these that involve ratios wind up having vertical asymptotes in their graphs. That is, places where the function is undefined and the denominator goes to zero. The inverse of the trigonometric functions are very useful, but somewhat treacherous. You've got to be careful. Some students like to write the inverse of sine as sine with a -1 superscript. That can lead to some confusion thinking that it is the cosecant or the reciprocal of sine. That is unfortunate notation and I encourage you to use instead arcsine to denote the inverse of the sine function. The arcsine is that function which, when composed with sine, gives you the identity back. One of the things that you'll note about both the arcsine and the arccosine is that they have a restricted domain. The domain must be the closed interval from negative 1 to 1, because of course, sine and cosine can only take values in that interval. In contrast to this, the arctangent function does have an infinite domain. Its range, however, is limited between negative pi over two and pi over two. These are all functions that you're going to want to be familiar with for moving forward in calculus. The last class of functions that is of critical importance are the exponential functions. These are functions of the form, e to the x, e to the x being the canonical example of an exponential function. Its inverse is the logarithm, or more precisely, the natural logarithm, ln of x. If you've seen these functions before, you know, because they're inverses, that their graphs have this symmetry about the diagonal. So, for example, if e to the zero is one, as it must be, then log of one must be equal to zero, certainly. The question that is often unanswered in pre-calculus or even elementary calculus courses is, what is e and why is it so important in this exponential function? Well, one can say that e is that value whose logarithm is equal to one. But since we defined the natural logarithm in terms of base e, that's a bit of circular reasoning. How do we reason about e? Well certainly, e is a number. It is a particular location on the real number line. It has a decimal expansion, but being irrational, it is a little bit hard to remember all of it. That doesn't really answer the question of what is e? Why is it so important? Before we get to the answer of that, let's review some of the algebraic properties associated with the exponential function. I hope you remember that e to the x times e to the y is e to the x+y. The exponents add. And e to the x raised to the y power is e to the x times y. In your prior exposure to calculus, I'm sure that you've seen some of the differential and integral properties of this function. e to the x has this wonderful property that it is its own derivative. And of course that it is its own integral, at least up to a constant. These facts are easy to remember but maybe not so easy to fully comprehend. There is one last ingredient that we're going to need before we go deep into exploring what e to the x means. This is something called Euler's Formula. This is simple looking. It states that e to the i times x = cos of x + i times sine of x. This is a wonderful formula, whatever it means. What does it mean? What is this i that is in here? I am sure that you have seen before the notation for the square root of negative 1, for the primal imaginary number, i. That is what is meant in Euler's Formula. It has the property, of course, that i squared is equal to -1. Well, this is a formula. It's a true statement. But what does it mean? Well, that is the subject for our next lesson. And so we end with a mystery about the exponential function and what it means to exponentiate something that is not a real number. In our next lesson, we'll resolve this mystery, in part, by contemplating what the exponential function is.
