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...