Page 110 - DMTH401_REAL ANALYSIS

Real Analysis

Notes The functions which are not rational are known as irrational functions. A typical example of an
irrational function is the square root function which we define as follows:

Definition 7: Square Root Function
Let S be the set of non-negative real numbers. A function f: S  R defined by

f(x) = x , " x  S
is called the square root function.

You may recall that x is the non-negative real number whose square is x. Also it is defined for
all x  0.
Polynomial functions, rational functions and the square root function are some of the examples
of what are known as algebraic functions. An algebraic function, in general, is defined as follows
Definition 8: Algebraic Function

An algebraic function f : S  R is a function defined by y = f(x) if it satisfies identically an
equation of the form
n–1
n
p (x)y + p (X)y +. . . . + p (x)y + p (x) = 0
0 1 n–1 n
where p(x), p (x), .... p (x), p (x) are Polynomials in x for all x in S and n is a positive integer.
l n–1 n
Example: Show that f: R  R defined by

2

x  3x 2
f(x) =
4x 1

is an algebraic function.
Solution:
2

x  3x 2
Let y = f(x) =
4x 1

2
Then (4 x – 1) y – (x – 3x + 2) = 0
2
Hence f(x) is an algebraic function.
In fact, any function constructed by a finite number of algebraic operations (addition, subtraction,
multiplication, division and root extraction) on the identity function and the constant function,
is an algebraic function.

Example: The functions f : R  R defined by

2

(x  2) x 1
(i) f(x) = 2
x  4
2
x  2x
or f(x) =
2
x.(3x  5)
are algebraic functions.

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