3.1 What Are Functions?

Functions are what we use to describe things we want to talk about mathematically. I find, though, that I get a bit tongue tied when I try to define them.

The simplest definition is: a function is a bunch of ordered pairs of things (in our case the things will be numbers, but they can be otherwise), with the property that the first members of the pairs are all different from one another.

Thus, here is an example of a function:

\[[\{1, 1\}, \{2, 1\}, \{3, 2\}]\]

This function consists of three pairs, whose first members are \(1, 2\) and \(3\).
It is customary to give functions names, like \(f, g\) or \(h\), and if we call this function \(f\), we generally use the following notation to describe it:

\[f(1) = 1, f(2) = 1, f(3) = 2\]

The first members of the pairs are called arguments and the whole set of them is called the domain of the function. Thus the arguments of \(f\) here are \(1, 2\) and \(3\), and the set consisting of these three numbers is its domain.

The second members of the pairs are called the values of the functions, and the set of these is called the range of the function.

The standard terminology for describing this function f is:

The value of \(f\) at argument \(1\) is \(1\), its value at argument \(2\) is \(1\), and its value at argument \(3\) is \(2\), which we write as \(f(1) = 1, f(2) = 1, f(3) = 2\).

We generally think of a function as a set of assignments of values (second members of our pairs) to arguments (their first members).

The condition that the first members of the pairs are all different is the condition that each argument in the domain of \(f\) is assigned a unique value in its range by any function.

Exercise 3.1 Consider the function \(g\), defined by the pairs \((1, 1), (2, 5), (3, 1)\) and \((4, 2)\). What is its domain? What is the value of \(g\) at argument \(3\)? What is \(g(4)\)?

If you stick a thermometer in your mouth, you can measure your temperature, at some particular time. You can define a function \(T\) or temperature, which assigns the temperature you measure to the time at which you remove the thermometer from your mouth. This is a typical function. Its arguments are times of measurement and its values are temperatures.

Of course your mouth has a temperature even when you don't measure it, and it has one at every instant of time and there are an infinite number of such instants.

This means that if you want to describe a function \(T\) whose value at any time t is the temperatures in your mouth at that time, you cannot really list all its pairs. There are an infinite number of possible arguments \(t\) and it would take you forever to list them.

Instead, we employ a trick to describe a function \(f\): we generally provide a rule which allows you, the reader, to choose any argument you like in \(f\)'s domain, and, by using the rule, to compute the value of your function at that argument. This rule is often called a formula for the function. The symbol \(x\) is often used to denote the argument you will select, and the formula tells you how to compute the function at that argument.

The simplest function of all, sometimes called the identity function, is the one that assigns as value the argument itself. If we denote this function as \(f\), it obeys

\[f(x) = x\]

for \(x\) in whatever domain we choose for it. In other words, both members of its pairs are the same wherever you choose to define it.

We can get more complicated functions by giving more complicated rules, (These rules are often called formulae as we have noted already). Thus we can define functions by giving any of the following formulae among an infinity of possibilities:

\[3x, x^2, x^2-1, \frac{3}{x}, x^3, \frac{x}{x^2 + 1}, 3x + 5, x^2 + 7x - 1\]

These represent, respectively, \(3\) times \(x\), \(x\) squared, \(x\) squared minus \(1\), \(3\) divided by \(x\), \(x\) cubed, \(x\) divided by the sum of the square of \(x\) and \(1\), and so on.

We can construct functions by applying the operations of addition, subtraction, multiplication and division to copies of \(x\) and numbers in any way we see fit to do so.

There are two very nice features of functions that we construct in this way, and the first applies to all functions.

We can draw a picture of a function, called its graph, on a piece of graph paper, or on a spreadsheet, chart, or with a graphing calculator. We can do it by taking argument-value pairs of the function and describing each by a point in the plane, with \(x\) coordinate given by the argument and y coordinate given by the value for its pair.

Of course it is impossible to plot all the pairs of a function that has an infinite domain, but we can get a pretty good idea of what its graph looks like by taking perhaps a hundred evenly spaced points in any interval of interest to us. This sounds like an impossibly tedious thing to do and it used to be so, but now it is not. On a spreadsheet, the main job is to enter the function once (with its argument given by the address of some other location). That and some copying is all you have to do, and with practice it can be done in \(30\) seconds for a very wide variety of functions.

The second nice feature is that we can enter any function formed by adding, subtracting, multiplying, dividing and performing still another operation, on the contents of some address very easily on a spreadsheet or graphing calculator. Not only that, these devices have some other built in functions that we can use as well.

The two of these facts mean that we can actually look at any function formed by adding subtracting multiplying or dividing copies of the identity function \(x\) and other built in functions, and any number we want, and see how they behave, with very limited effort.

We will soon see that we can use the same procedure used for graphing functions to graph their derivatives (we have not defined these yet) as well, but that is getting ahead of the story. You should realize though that we can compute derivatives for most functions numerically with only a small amount of effort as well.

Exercise 3.2 What is the value of the function \(x^2 + 3\) at \(x = 5\)? At argument \(10\)?

Would you please give some examples?