Page 7 - Module 1 in MATH 1 (Calculus)

Lesson Proper :

A. Definition of Function: A function is a rule that associates, with each value of a variable in
a certain set, exactly one value of another variable . The variable is then called the dependent
variable, and is called the independent variable. The set from which the values of can be
chosen is called the domain of the function. The set of all the corresponding values of is called
the range of the function.

EXAMPLE 1: The equation − = , with the independent variable, associates one

value of y with each value of x. The function can be calculated with the formula = − . The

domain is the set of all real numbers. The same equation, − = , with taken as the
independent variable, sometimes associates two values of with each value of . Thus, we must
distinguish two functions of : = √ + and = −√ + . The domain of both these functions
is the set of all y such that + ≥ or equivalent to ≥ − , since √ + is not a real number
when + < .
(Note :)
If a function is denoted by a symbol , then the expression ( ) denotes the value obtained
When is applied to a number in the domain of . Often, a function is defined by giving the

formula for an arbitrary value ( ) . For example, the formula ( ) = − determines the
first function mentioned in Example 1. The same function also can be defined by an equation like

= −

EXAMPLE 2:

(a) If ( ) = − + 2, then
3
(1) = (1) − 4(1) + 2 = 1 − 4 + 2 = −1
3
(−2) = (−2) − 4(−2) + 2 = − 8 + 8 + 2 = 2
3
( ) = − 4 + 2

(b) The function ( ) = − is defined for every number ; that is, without exception,

− is a real number whenever is a real number. Thus, the domain of the function is
the set of all real numbers.

(c) The area of a certain rectangle, one of whose sides has length , is given by = − .

Here, both and must be positive. By completing the square, we obtain = − ( − ) +

27. In order to have > 0, we must have ( − ) < 27, which limits to values below 6;
hence, < < . Thus, the function determining has the open interval ( , ) as domain.
From the (1), we see that the range of the function is the interval (0,27].
Notice that the function of part (c) here is given by the same formula as the function of part (b), but the
domain of the former is a proper subset of the domain of the latter

Figure 1.

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