Let's start with a picture. This is a picture of a black box. We can tell this black box to do anything, but because this section is about functions, this black box is going to do calculations with numbers. Suppose we decide to have this black box "take a number, multiply it by 3 and then subtract 2."



It should be quite clear what happens as each number is put into the black box. Let's say a "4" is entered. The black box will take the "4", multiply it by 3, subtract 2 and out will come a "10".

In similar fashion, if a "2" had been put in, the output would have been "4".

The values that can be put into this black box are referred to as the domain. The domain is the set of all possible values that can be used as input. Sometimes we may wish to restrict the domain to only certain numbers. We do this by writing it like so: D = {1,2,5}. This means that the domain is restricted so that only the three numbers 1, 2 and 5 could be input into the black box.

The values that come out of the black box are referred to as the range. The range is the set of all possible values that are output. Using our restricted domain, then the range is the set of numbers 1, 4 and 13, or R = {1,4,13}

The black box follows a particular rule: "takes a number, multiplies it by 3 and subtracts 2."

There are three parts to what is going on: the "domain", the "rule" and the "range".

Now for a bit more terminology. The domain, often denoted by x, is the independent variable, whereas the range, often denoted as y, is the dependent variable. Once we select an x, such as a 5, to put into the black box, the output, or the y-value is dependent upon it.

A function is a rule that assigns to each element in the domain one and only one element in the range. -- The Definition of a Function

Notice that the definition refers to three parts: domain, rule and range. The key concept, though, is the one and only one phrase. For each value put in, only a single value may come out if the expression is to be a function. Otherwise the expression is classified as a relation.

Now, we can combine all the above information into a single statement using functional notation.

y = f(x) = 3x - 2 with D = {1,2,5} [Equation A]

The above statement is read "y equals f of x equals 3 x minus 2 with domain D, equal to the set of numbers 1, 2 and 5.

Caution: f(x) is read "f of x" and does not mean multiplication. It is used to indicate that the name of the function is f and that the variable inside the parentheses, in this example x, is to be the independent variable.

It is time to stop and try some questions. Using f as defined in Equation A , answer the following questions.

Question 1. If x = 2, what is y?
Question 2. If x = 5, what is y?
Question 3. If x = 7, what is y?

Answer 1. y = 3*2 - 2 = 6 - 2 = 4
Answer 2. y = 13
Answer 3. y is undefined, because 7 is not an element of the domain. f is defined only for the numbers 1, 2 and 5, which was an arbitrary decision on my part.

That probably wasn't so hard. The confusing part for people comes when the exact same questions are asked using functional notation. Here are some examples. Continue to use f as in Equation A.

Question 4. Find f(2)
Question 5. Find f(5)
Question 6. Find f(7)


Answer 4. f(2) read "f of 2" is asking, If x = 2, what is y?. It is exactly the same as question 1 above, and the answer is 4, or f(2) = 4.
Answer 5. f(5) is asking the same thing as question 2 above, but using function notation. The question is asking, If the independent variable is 5 and the rule is f, what is the value of the dependent variable? As in question 2 above, the answer is 13, or f(5) = 13.
Answer 6. f(7) is undefined, because the independent variable 7 is not an element of the domain.