Page 108 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Note that both f(x) and g(x) are elements of R. Hence each of their sum and difference is again a
unique member of R.
Definition 2: Product of Two Functions

Let f: S R and g: S  R be any two functions. The product of f and g, denoted as f, g, is defined
as a function f. g: S  R by
(f . g) (x) = f(x) . g(x), " x S.

Definition 3: Scalar Multiple of a Function
Let f: S  R be a function and k be same fixed real number. Then the scalar multiple of ‘f’ is a
function k f S  R defined by

(kf (x) = k. f(x), " x  S.
This is also called the scalar multiplication.
Definition 4: Quotient of Two Functions
Let f: S  R and g: S  R be any two functions such that g(x)  0 for each x in S. Then s function
f
: S  R defined by
g
 f  f(x)
  (x) = , " x  S
g
  g(x)
is called the quotient of the two functions.
2
Exercise 1: Let f, g, h be any three functions, defined on S and taking values in R, as f (x) = ax ,
g(x) = bx for every x in S, where a, b, are fixed real numbers. Find f + g, f – g, f, g, f/g and kf,
when k is a constant.

9.1.2 Notion of an Algebraic Function

2
You are quite familiar with the equations ax + b = 0 and ax + bx + c = 0, where a, b, c  R,
a  0. These equations, as you know are, called linear (or first degree) and quadratic (or second
degree) equations, respectively. The expressions ax + b and ax + bx + c are, respectively, called
2
the first and second degree polynomials in x. In the same way an expression of the form ax +
2
bx + cx + d (a  0, a, b, c, d ER) is called a third degree polynomial (cubic polynomial) in x. In
n
general, an expression of the form a x + a x + a x + .... + a where a  0, a ER, i = 0, 1, 2,
n–l
n–2
o 1 2 n 0
...., n, is called an nth degree polynomial in x.
A function which is expressed in the form of such a polynomial is called a polynomial function.
Thus, we have the following definition:
Definition 5: Polynomial Function
Let a (i = 0, 1, ...., n) be fixed real numbers where n is some fixed non-negative integer. Let S be
1
a subset of R. A function f: S  R defined by
a–1
n–2
n
f(x) = a x + a x + a x + .... + a , " x S, a  0.
0 1 2 n 0
is called a polynomial function of degree n.
Let us consider some particular cases of a polynomial function on R:
Suppose f : S  R is such that
(i) f(x) = k, " x  S (k is a fixed real number). This is a polynomial function. This is generally
called a constant function on S.

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