Applied Mathematics for Business and Economics

Lecture Note Function

Chapter 1

Functions

1 Definition of a Function

1.1 Definition ..................................................................................................

Let D and R be two sets of real numbers. A function f is a rule that matches each
number x in D with exactly one and only one number y or f(x) inR. D is called the

domain of f and R is called the range of f. The letter x is sometimes referred to as
independent variable and y dependent variable.

Examples 1:

Letf.^23 32)( xxxx ++−= 100 Findf( (^2) ).
Solution:
(^) f(2 =2)^32 −× +×+ =2 2 3 2 100 106
Examples 2
A real estate broker charges a commission of 6% on Sales valued up to $300,000. For
sales valued at more than $ 300,000, the commission is $ 6,000 plus 4% of the sales
price.
a. Represent the commission earned as a function R.
b. Find R (200,000).
c. Find R (500,000).
Solution
a. ()
0.06 for 0 300, 000
0.04 6000 for 300, 000
xx
Rx
xx

⎧ ≤≤

=⎨

⎩ +>

b. Use R()xx=0.06 since 200, 000 300, 000<

(^) R()200, 000 =× =0.06 200, 000 $12, 000
c. Use Rx()=+0.04x 6000 since 50 0, 000 300, 000>
(^) R()500, 000 =× + =0.04 500, 000 6000 $26, 000

1.2 Domain of a Function ...............................................................................

The set of values of the independent variables for which a function can be evaluated is
called the domain of the function.

Dx=∈ ∃∈ ={ \\/,yyf(x)}^

Example 3
Find the domain of each of the following functions:

a. ()

1

3

fx

x

=

−

, b. gx( )= x−^2

Solution
a. Since division by any real number except zero is possible, the only value of x
for which ()

1

3

fx

x

=

−

cannot be evaluated isx= 3 , the value that makes the

denominator of f equal to zero, or D=−\ { (^3) }.