The concept of a function is more central in mathematics than the concept of a number. In this class we use functions only in a very narrow context, but if you go on in mathematics you will encounter functions ubiquitously in every class you take.
For the purposes of this class, you can think of a function as a machine that takes a number (the input), processes it, and produces another number (the output). A given input always produces the same output, but the same output can be produced by different inputs. The way a particular function processes the input is usually described by an algebraic expression (sometimes called the rule). Processing a particular input is called evaluating the function at . The set of all numbers for which you can in fact evaluate the function is called the domain of the function, the set of all possible outputs you obtain in this manner is called the range of the function.
For example, consider the equation
Sometimes you will be asked to determine the domain of a function. There are more subtle situations, but in this class the domain is always the set of all real numbers, except those where you cannot evaluate the expression. Usually the only reason you might be unable to evaluate a function is that the relevant expression might call for a division by zero, or the computation of a square root of a negative number.
For example, the domain of the function
It is easy to be confused about just what it means to evaluate a function. Functions can be evaluated not just at numbers, but also at algebraic expressions, and at other function values. Let's look at some examples. Suppose that is defined
Functions can be combined in various ways to create new functions. Suppose and are two functions, and is one of the arithmetic operations , , , or . Then a new function
For example, suppose as before that
A function can be evaluated at the value of another (or the same) function. This is called the composition of functions. The composition of two functions and is denoted by
Note that, on the other hand,
There are subtle issues regarding the domain and range of the
functions involved. In particular, when we consider a composition
like the range of must be a subset of the domain of
The graph of a function is the
graph of the equation
Graph of a function.
The graph of a function is the graph of the equation