Functions (§3.1)

Functions are an important type of mathematical object, along with relations, sets, and constant numbers. These notes show functions in context with these other kinds of things.

There are many different types of mathematical objects that we could study in this class. Some of them are relation-like objects:

truth values,
sets,
relations,
ternary relations,
etc;

some of them are function-like objects:

constants,
functions,
binary functions,
ternary functions,
etc.

As you go along these lists, both the number of variables and the number of dimensions needed for graphing increase, as in the following diagram:

A truth value is either true or false; any statement with no variables in it, such as the statement that 0 < 2, should evaluate to true or false (in this case, true). To indicate that you are talking about the truth value of this statement, rather than asserting the statement itself, you can put curly braces around it (although there are several other notations used for this); for example, {0 < 2} is the truth value that 0 is less than 2, which is the true truth value rather than the false one. The graph of the true truth value is a solid dot, while the graph of the false truth value is a hollow circle; either way, this takes zero dimensions. You can also use a variable to give a name to a truth value, so maybe p stands for {0 < 2}; we won't need to do that in this course, but you'll do it constantly if you take a course on Logic.

A constant is, in this class, usually a real number, such as −2. Any expression with no variables should evaluate to a constant (possibly undefined), and we use one dimension to graph a constant on a number line. Again, you can use a variable to stand for a constant, so maybe a stands for −2; in other words, a = −2.

A set is, in the simplest case, a set of real numbers. A statement with one variable defines a set, such as {x | x < 2}, the set of real numbers that are less than 2. We again use one dimension to graph a set. If A stands for the set {x | x < 2}, then these two statements mean the same thing:

x ∈ A, usually pronounced ‘x in A’;
x < 2.

The first of these says that x belongs to the set A, while the second uses the definition of A to say exactly what that means about x.

A function, or unary function for emphasis, is a rule for taking a number (the input) and using it to calculate a number (the output). An example is (x ↦ x − 2), the rule which subtracts 2 from any number. To graph a function, we need two dimensions, one for the input and one for the output. If f stands for the function (x ↦ x − 2), then these two expressions mean the same thing:

f(x), usually pronounced ‘f of x’;
x − 2.

The first of these is the value of the function f at the argument x, while the second uses the definition of f to say exactly what that means in terms of x.

A relation, or binary relation for emphasis, is a set of ordered pairs instead of a set of individual numbers. An example is {x, y | x + y < 2}. We again use two dimensions to graph a relation.

We can continue with binary functions, ternary functions, etc, which take two or more numbers as inputs; and we can continue with ternary relations, quaternary relations, etc, which relate three or more numbers. But we will not actually study these in this class.

One of the basic principles of the theory of functions is that the only information necessary to specify a function is to show how to calculate its value at any argument. Therefore, if I write

f(x) = x − 2

and state that this holds for every real number x, then I have completely specified the function f. Usually in class (and always in the book), we will define a function with a name like this, but really it's just another way of saying that f is the function (x ↦ x − 2). It's a handy method, because you have an equation in which you can replace x with any other expression (since the equation holds for every real number x). For example, using the function f above, we can replace x with 5 to calculate that

f(5) = (5) − 2 = 3.

Or we can say that

f(2x + 3) = (2x + 3) − 2 = 2x + 1.

Notice that I always put parentheses around an expression when I substitute it for a variable; in this case, it wasn't really necessary, but it's best to play it safe.

Every function can be thought of as a relation, specifically

{x, y | y = f(x)}.

Conversely, if a relation is given by an equation in x and y and you can solve this equation uniquely for y, then the relation can be thought of as a function. In the graph, this corresponds to the Vertical Line Test:

A graph of a relation is the graph of a function if and only if every vertical line goes through the graph at most once.

Sometimes a vertical line doesn't go through the graph at all! This happens when f(x) is undefined. The domain of f is the set of all inputs where f is defined:

dom f = {x | f(x) exists}.

Along with the domain, we can also consider the range, which is the set of all outputs of the function; however, calculating that from a formula will be easier after we discuss inverse functions (§3.7), or using Calculus (but we won't do that in this course).

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