Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.3

Algebraic function : A function in the form of a polynomial with finite number of terms is known as algebraic
function.

Example 11 : x 2x 3, x 1etc.^2 + −^2 +

Domain and Range of a Function :
The set of values of independent variables x is called the ‘Domain’ of the function and the set of
corresponding values of f(x) i.e. the dependent variable y is called the ‘Range’ of the function.
Example 12 : For the squared function of y = x^2 , we get the ordered pairs (1, 1) (2, 4) (3, 9), ...., The set of
independent variables {... –2, –1, 0, 1, 2, 3, ...} is the domain, where as set of dependent variable {0, 1, 4,
9, ....} represents the range.
Example 13 : For the following functions find the domain and range.

(i) ( )
x 4^2
f x = x 2−− , x 2≠ (ii) f x( )=3 xx 3−− , x 3≠ (iii) ( ) ( ) ( )
f x 2x 1
x 1 x 1

= +

− +

Solution:

(i) f 0( )=−−^42 =2, f 1( )=−−^31 =3, f 3( )=^51 =5, f 4 6, f 1 1( )= ( )− =

∴ domain = { –1, 0, 1, 3, 4, .....}, range = {1, 2, 3, 5, 6, ....}

= R – {2}, = R – {4}, R = real number,

(ii) f 0( )=−^33 = −1, f 1 1, f 2 1, f 1 1( )= − ( )= − ( )− = −

domain = { – 1, 0, 1,2, 4, ....}, range = { –1, –1, –1, ....} = {–1}
= R – {3}, where R is a real number
(iii) The value of f(x) exists for x ≠ 1, x ≠ – 1.
∴ domain = R – {– 1, 1}, i.e, all real numbers excluding 1 & (–1)
range = R.

Example 14 : Find the domain of definition of the function 2
4 x 5
x 7x 12

−

− +

Denominator = x 7x 12^2 − + = (x 3 x 4 .− )( − ) Given function cannot be defined for 3 ≤ x ≤ 4. So domain is
for all real values of x except 3 and 4.

Absolute Value : A real number “a” may be either a = 0, or a > 0 or, a < 0. The absolute value (for modulus)
of a, denoted by | a | is defined as | a | = a, for a > 0
= – a, for a < 0
Thus | – 4| = – (– 4) = 4, and | 4 | = 4.
Complex No:
A number of the form (a+ib) or (a–ib); where a & b are real numbers is called a complex number (where
i 1= − )
The complex number has two parts; a real part & an imaginary part. ‘a’ is the real part & ‘ib’ is the
imaginary part.