WHAT IS A FUNCTION?
No concept in mathematics, especially in calculus, is more fundamental than the concept of a function. The term was first used in a 1673 letter written by Gottfried Wilhelm Leibniz, the German mathematician and philosopher who invented calculus independently of Isaac Newton. Since then the term has undergone a gradual extension of meaning.
In traditional calculus a function is defined as a relation between two terms called variables because their values vary. Call the terms x and y. If every value of x is associated with exactly one value of y, then y is said to be a function of x. It is customary to use x for what is called the independent variable, and y for what is called the dependent variable because its value depends on the value of x.
As Thompson explains in Chapter 3, letters at the end of the alphabet are traditionally applied to variables, and letters elsewhere in the alphabet (usually first letters such as a,b,c ...) are applied to constants. Constants are terms in an equation that have a fixed value. For example, in y = ax + b, the variables are x and y, and a and b are constants. If y = 2x + 7, the constants are 2 and 7. They remain the same as x and y vary.
A simple instance of a geometrical function is the dependence of a square's area on the length of its side. In this case the function is called a one-to-one function because the dependency goes both ways. A square's side is also a function of its area.
A square's area is the length of its side multiplied by itself. To express the area as a function of the side, let y be the area, x the side, then write y = x2 It is assumed, of course, that x and y are positive.
A slightly more complicated example of a one-to-one function is the relation of a square's side to its diagonal. A square's diagonal is the hypotenuse of an isosceles right triangle. We know from the Pythagorean theorem that the square of the hypotenuse equals the sum of the squares of the other two sides. In this case the sides are equal. To express the diagonal as a function of the square's side, let y be the diagonal, x the side, and write y=[square root of 2 x2] or more simply y=x [square root of 2] to express the side as a function of the diagonal, let y be the side, x the diagonal, and write y=[square root of 2 x2/2], or more simply y= x/[square root of 2] .
The most common way to denote a function is to replace y, the dependent variable, by f(x) — f being the first letter of "function." Thus y = f(x) = x2 means that y, the dependent variable, is the square of x. Instead of, say, y = 2x – 7, we write y = f(x) = 2x – 7. This means that y, a function of x, depends on the value of x in the expression 2x – 7. In this form the expression is called an explicit function of x. If the equation has the equivalent form of 2x – y – 7 = 0, it is called an implicit function of x because the explicit form is implied by the equation. It is easily obtained from the equation by rearranging terms. Instead of f(x), other symbols are often used.
If we wish to give numerical values to x and y in the example y = f(x) = 2x – 7, we replace x by any value, say 6, and write y = f(6) = (2 · 6) – 7, giving the dependent variable y a value of 5.
If the dependent variable is a function of a single independent variable, the function is called a function of one variable. Familiar examples, all one-to-one functions, are:
The circumference or area of a circle in relation to its radius.
The surface or volume of a sphere in relation to its radius.
The log of a number in relation to the number.
Sines, cosines, tangents, and secants are called trigonometric functions. Logs are logarithmic functions. Exponential functions are functions in which x, the independent variable, is an exponent in a equation, such as y = 2 There are, of course, endless other examples of more complicated one-variable functions which have been given names.
Functions can depend on more than one variable. Again, there are endless examples. The hypotenuse of a right triangle depends on its two sides, not necessarily equal. (The function of course involves three variables, but it is called a two-variable function because it has two independent variables.) If z is the hypotenuse, we know from the Pythagorean theorem that z=[square root of x2+y2. Note that this is not a one-to-one function. Knowing x and y gives z a unique value, but knowing z does not yield unique values for x and y.
Two other familiar examples of a two-variable function, neither one-to-one, are the area of a triangle as a function of its altitude and base, and the area of a right circular cylinder as a function of its radius and height.
Functions of one and two variables are ubiquitous in physics. The period of a pendulum is a function of its length. The distance covered by a dropped stone and its velocity are each functions of the elapsed time since it was dropped. Atmospheric pressure is a function of altitude. A bullet's energy is a two-variable function dependent on its mass and velocity. The electrical resistance of a wire depends on the length of the wire and the diameter of its circular cross section.
Functions can have any number of independent variables. A simple instance of a three-variable function is the volume of a rectangular room. It is dependent on the room's two sides and height. The volume of a four-dimensional hyper-room is a function of four variables.
A beginning student of calculus must be familiar with how equations with two variables can be modeled by curves on the Cartesian plane. (The plane is named after the French mathematician and philosopher René Descartes who invented it.) Values of the independent variable are represented by points along the horizontal x axis. Values of the dependent variable are represented by points along the vertical y axis. Points on the plane signify an ordered pair of x and y numbers. If a function is linear — that is, if it has one form y = ax + b — the curve representing the ordered pairs is a straight line. If the function does not have the form ax + b the curve is not a straight line.
Figure 1 is a Cartesian graph of y = x.2 The curve is a parabola. Points along each axis represent real numbers (rational and irrational), positive on the right side of the x axis, negative on the left; positive at the top of the y axis, negative at the bottom. The graph's origin point, where the axes intersect, represents zero. If x is the side of a square, we assume it is neither zero nor negative, so the relevant curve would be only the right side of the parabola. Assume the square's side is 3. Move vertically up from 3 on the x axis to the curve, then go left to the y axis where you find that the square of 3 is 9. (I apologize to readers for whom all this is old hat.)
If a function involves three independent variables, the Cartesian graph must be extended to a three-dimensional space with axes x, y, and z. I once heard about a professor, whose name I no longer recall, who liked to dramatize this space to his students by running back and forth while he exclaimed "This is the x axis!" He then ran up and down the center aisle shouting "This is the y axis!", and finally hopped up and down while shouting "This is the z axis!" Functions of more than three variables require a Cartesian space with more than three axes. Unfortunately, a professor cannot dramatize axes higher than three by running or jumping.
Note the labels "domain" and "range" in Figure 1. In recent decades it has become fashionable to generalize the definition of function. Values that can be taken by the independent variable are called the variable's domain. Values that can be taken by the dependent variable are called the range. On the Cartesian plane the domain consists of numbers along the horizontal (x) axis. The range consists of numbers along the vertical (y) axis.
Domains and ranges can be infinite sets, such as the set of real numbers, or the set of integers; or either one can be a finite set such as a portion of real numbers. The numbers on a thermometer, for instance, represent a finite interval of real numbers. If used to measure the temperature of water, the numbers represent an interval between the temperatures at which water freezes and boils. Here the height of the mercury column relative to the water's temperature is a one-to-one function of one variable.
In modern set theory this way of defining a function can be extended to completely arbitrary sets of numbers for a function that is described not by an equation but by a set of rules. The simplest way to specify the rules is by a table. For example, the table in Figure 2 shows a set of arbitrary numbers that constitute the domain on the left. The corresponding set of arbitrary numbers in the range is on the right. The rules that govern this function are indicated by arrows. These arrows show that every number in the domain correlates to a single number on the right. As you can see, more than one number on the left can lead to the same number on the right, but not vice versa. Another example of such a function is shown in Figure 3, along with its graph, consisting of 6 isolated points in the plane.
Because every number on the left leads to exactly one number on the right, we can say that the numbers on the right are a function of those on the left. Some writers call the numbers on the right "images" of those on the left. The arrows are said to furnish a "mapping" of domain to range. Some call the arrows "correspondence rules" that define the function.
For most of the functions encountered in calculus, the domain consists of a single interval of real numbers. The domain might be the entire x axis, as it is for the function y = x.2 Or it might be an interval that's bounded; for example, the domain of y = arcsin x consists of all x such that –1 =[less than or equal to] x ≤1. Or it might be bounded on one side and unbounded on the other; for example, the domain y=[square root of x]consists of all x [greater than or equal to 0]. We call such a function "continuous" if its graph can be drawn without lifting the pencil from the paper, and "discontinuous" otherwise. (The complete definition of continuity, which is also applicable to functions with more complicated domains, is beyond the scope of this book.)
For example, the three functions just mentioned are all continuous. Figure 4 shows an example of a discontinuous function. Its domain consists of all real numbers, but its graph has infinitely many pieces that aren't connected to each other. In this book we will be concerned almost entirely with continuous functions.
Note that if a vertical line from the x axis intersects more than one point on a curve, the curve cannot represent a function because it maps an x number to more than one y number. Figure 5 is a graph that clearly is not a function because vertical lines, such as the one shown dotted, intersect the graph at three spots. (It should be noted that Thompson did not use the modern definition of "function." For example the graph shown in Figure 30 of Chapter XI fails this vertical line test, but Thompson considers it a function.)
In this generalized definition of function, a one-variable function is any set of ordered pairs of numbers such that every number in one set is paired with exactly one number of the other set. Put differently, in the ordered pairs no x number can be repeated though a y number can be.
In this broad way of viewing functions, the arbitrary combination of a safe or the sequence of buttons to be pushed to open a door, are functions of counting numbers. To open a safe you must turn the knob back and forth to a random set of integers. If the safe's combination is, say, 2-19-3-2-19, then those numbers are a function of 1,2,3,4,5. They represent the order in which numbers must be taken to open the safe, or the order in which buttons must be pushed to open a door. In a similar way the heights of the tiny "peaks" along a cylinder lock's key are an arbitrary function of positions along the key's length.
In recent years mathematicians have widened the notion of function even further to include things that are not numbers. Indeed, they can be anything at all that are elements of a set. A function is simply the correlation of each element in one set to exactly one element of another set. This leads to all sorts of uses of the word function that seem absurd. If Smith has red hair, Jones has black hair, and Robinson's hair is white, the hair color is a function of the three men. Positions of towns on a map are a function of their positions on the earth. The number of toes in a normal family is a function of the number of persons in the family. Different persons can have the same mother, but no person has more than one mother. This allows one to say that mothers are a function of persons. Elephant mothers are a function of elephants, but not grandmothers because an elephant can have two grandmothers. As one mathematician recently put it, functions have been generalized "up to the sky and down into the ground."
A useful way to think of functions in this generalized way is to imagine a black box with input and output openings. Any element in a domain, numbers or otherwise, is put into the box. Out will pop a single element in the range. The machinery inside the box magically provides the correlations by using whatever correspondence rules govern the function. In calculus the inputs and outputs are almost always real numbers, and the machinery in the black box operates on rules provided by equations.
Because the generalized definition of a function leads to bizarre extremes, many educators today, especially those with engineering backgrounds, think it is confusing and unnecessary to introduce such a broad definition of functions to beginning calculus students. Nevertheless, an increasing number of modern calculus textbooks spend many pages on the generalized definition. Their authors believe that defining a function as a mapping of elements from any set to any other set is a strong unifying concept that should be taught to all calculus students.
Opponents of this practice think that calculus should not be concerned with toes, towns, mothers, and elephants. Its domains and ranges should be confined, as they have always been, to real numbers whose functions describe continuous change.
It is a fortunate and astonishing fact that the fundamental laws of our fantastic fidgety universe are based on relatively simple equations. If it were otherwise, we surely would know less than we know now about how our universe behaves, and Newton and Leibniz would probably never have invented (or discovered?) calculus.CHAPTER 2
WHAT IS A LIMIT?
It is possible, though difficult, to understand calculus without a firm grasp on the meaning of a limit. A derivative, the fundamental concept of differential calculus, is a limit. An integral, the fundamental concept of integral calculus, is a limit.
To explain what is meant by a limit, we will be concerned in this chapter only with limits of discrete functions because limits are easier to understand in discrete terms. When you read Calculus Made Easy you will learn how the limit concept applies to what are called functions of a continuous variable because their variables have real number values that vary continuously. Functions of discrete variables have variables whose values jump from one value to another. There are also functions of complex variables in which the values are complex numbers — numbers based on the imaginary square root of minus one. Complex variables are outside the scope of Thompson's book.
A sequence is a set of numbers in some order. The numbers don't have to be different and they need not be integers. Consider the sequence 1,2,3,4,. ... This is just the positive integers. It is an infinite sequence because it continues without stopping. If it stopped it would be a finite sequence.