Functions.html

FUNCTIONS

Objectives: To recall the concept of a function.

Recall 1: (Definition of a function.)

Suppose that A and B denote two sets of objects. They may be sets of numbers, sets of books, sets of humans, etc. Then a rule that assigns exactly one member of the set B to each member of the set A is called a function from A to B. For example, if A represents the set of books, and B represents the set of whole numbers, then the rule which assigns to every book, the number of pages in that book is a function from the set of books to the set of whole numbers.

Although the function concept is general enough to be defined for arbitrary sets A and B as described above, in mathematics it is normally used when both A and B are sets of numbers. We will apply the function concept to the situation where both A and B are the set of real numbers.

Note
We use the word function in mathematics exactly like we do in english. In english, we often say things like "His mood is a function of the weather," or "His grades are a function of his work habits." In these statements one could replace the words a function by the word dependent on. In our example above we could say, "the number of pages of a book is a function of the book," or "the number of pages of a book depends on the book."

Recall 2: (The domain of a function)

The domain of a function from A to B is the members of A that have corresponding members of B that are assigned by the rule that the function describes. In the book example above, the domain is all books. However, the domain may not always be the whole set A. Take, for example, the function from the set of real numbers the square root of that number. That is,

etc.

The domain of this function is not the set of all real numbers because the negative numbers cannot be assigned a value. The square root of a negative number does not exist! (That is if we don't count the complex numbers!). In this case, the domain is the set of all non negative numbers.

Recall 3: (The range of a function)

The range of a function from A to B is the set of members of B that are assigned to members of A by the function. In the case of the books, the range is the numbers that are numbers of pages of books. Negative numbers are not in the range of this function, because a book cannot have a negative number of pages. Zero is not in the range, because a book with 0 pages is not considered a book (or is it?) Moreover, not all the positive whole numbers are in the range, because it may be that there are no books with (say) 13 pages; and, it is hardly likely that there are books with more than (say) 100000 pages.

Recall 4: (Independent and dependent variables)

When the sets A and B are sets of numbers (as they will be in most cases,) the members of the domain are called the independent variables while the members of the range are called the dependent variables. This makes sense, given the fact that the members of B depend on the members of A.

Recall 5: (Notation)

There are several forms of notation used in illustrating particular functions. Most common is the single letter notation like f or g We denote the function as f(x), meaning that the function f assigns the value f(x) to the value x. For example, if f is the squaring function (the function that takes each real number to its square), then it is denoted f(x) = x², meaning

   f(-2) = (-2)² = 4,
   f(2) = 2² = 4,
   f(3) = 3² = 9,
   f(4) = 4² = 16

Another notation that is commonly used in higher mathematics is f:A-> B, which is a more descriptive illustration, suggesting that f takes the values of A and sends them to values of B.

Recall 6: (How certain equations may be thought of as functions.)

Through most of the previous topics, we have worked with equations that involve only one variable. Equations like
x^2 - x + 40 = 0

These equations are useful in solving many problems. The word problems are examples of the usefulness of such equations. Many problems will involve more than one variable and it will become increasingly important to know how to treat equations of more than one variable, say an x and a y. Some of these equations may be put in the form where the y can be isolated and put on one side of the equation, with the remaining terms on the other side of the equation. Such is the case with the equation

In this case, if we may assume that x is different from 0, the equation can be put in the form y equals

This is an equation that may also be viewed as a function. Namely, the function that takes real numbers to real numbers in the following way:

    Take the real number x,
    square it,
    subtract 60
    and divide the result by 2 times x.

Let y be the number that you get from this recipe. Mathematically, it is clear that y must be the same as

Recall 7: (How the function concept makes equations more dynamic.) From Recall 6 an equation like

can be thought of as a function from the set of real numbers to the set of real numbers and this function may be denoted as

f(x) =
You may not think that there is much difference between thinking of the equation

and the function f(x) = ; but, the fact is that with the function concept we may find f(x+2), which is

This will become extremely useful when we learn how to graph functions.

Examples

Example 1. Find the domain of the following function that takes real numbers to real numbers.

f(x) = .

Solution: Pick any number x in the interval where -5 < x < 5. This function cannot handle such numbers because the expression under the root sign is negative for such numbers; and when we take the square root, to find the value that the function takes x to, we find that we are are attempting to computethe square root of a negative number. The square root of a negative number is not a real number --there is no real number whose square is negative. Therefore, we must exclude the real number between -5 and 5 from the domain. The natural domain of this function is all real numbers greater than or equal to 5 and all real numbers less than or equal to -5.

Example 2. Find the domain of the function.

f(x) =3x/[(x-3)(x+2)].

Solution: Notice that when x = 3 or when x = -2, the rational expression is not defined as a finite real number (they are poles). Since functions must take real numbers to real numbers (and since every real number must be finite), we must exclude the values x = 3 and x = -2 from the domain. So the domain includes all real numbers except -2 and 3.

Example 3. Find the range of the function

f(x) =(x + 3)² + 4

Solution: If x = -3, the term (x + 3)² vanishes. Also this term can never be less than zero. So f(x) is smallest when x = -3 and the value of f(-3) is 4. As x takes values greater than -3, the function takes values that are greater than f(-3). In fact any value greater than 4 is achieved by some value different from x = -3. Take any large value, say 10,000, and ask if there is some value of x where f(x) = 10,000. This amounts to solving the equation

10,000 = (x + 3)² + 4.

We may solve this equation by subtracting 4 from both sides to get

9,996 = (x + 3)².

This means that x + 3 = SQRT(9,996),
or,
x = -3 +SQRT(9,996).

We now know that f(-3+SQRT(9,996)) = 10,000.

Clearly, we may play the same game with any number greater than 4. So, the range of this function is the set of all real numbers greater than or equal to 4.

Example 4. Let f(x) be

Find f(x + 1).

Solution: The function f itself is the rule which says that if you give it a real number x then it will first
cube that number, add 2 times the square of that number, subtract 13 times that number and add 10,
then, divide the result by that number minus 1.

To find f(x + 1) simply follow the rules replacing that number by x+1. So, we have f(x+1) given by