Page 109 - DMTH401_REAL ANALYSIS

Unit 9: Functions

Notes
Example:

f(x) = 2, f(x) = – 3, f(x)= , " x  R, are all constant functions.
(ii) One special case of a constant function is, obtained by taking
k = 0 i.e. when
f(x) = 0, " x  S.

This is called the zero function on S.
Let f: S  R be such that
(iii) f(x) = a x + a , " x  S, a,  0.
0 1
This is a polynomial function and is called a linear function on S. For example,
f(x) = 2x + 3, f(x) = – 2 x + 3,
f(x) = 2x – 3, f(x) = –2x – 3, f(x) = 2x for every
x  S are all linear functions
(iv) The function f: S  R defined by

f(x) = x, " x  S
s called the identity function on S,
(v) f: S  R given as .

2
f(x),=a .x + a x + a , " x  R, a # O.
0 1 2 o
is a polynomial function of degree two and is called a quadratic function on S.

Example: f(x) = 2x + 3x – 4, f(x) = x + 3, f(x) = x + 2x,
2
2
2
f(x) = – 3x ,
2
for every x  S are all quadratic functions.
Definition 6: Rational Function
A function which can be expressed as the quotient of two polynomial functions is called a
rational function.
Thus a function f: S  R defined by


n
a x  a x n 1  ... a

f(x)  0 1 n , " x  S

m
b x  b x m 1  ...  b m
1
0
is called a rational function.
Here a  b  0, a , b  R where i, j are some fixed real numbers and the polynomial function in
0 0 i j
the denominator is never zero.
Example: The following are all rational functions on R.
2
2x  3 4x  3x  1 4 3x  5
, (x  ) and (x  4).
2
x  1 3x  4 3 x  4

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