Real Valued Functions

GREATEST INTEGER FUNCTION:

For any real numbers x, we denote [x], the greatest integer less than or equal to x

For example, [2.45] = 2, [-2.1] = -3, [1.75] = 1, [0.32] = 0 etc.

The function f  defined by f (x) = [x] for all x ϵ R, is called the greatest integer function.

In general, if n is an integer and x is any number satisfying n ≤ x < n+1, then [x] = n.

Also, if [x] denotes the fractional part of x, then [x] = x- [x] or x = [x] + [x].

PROPERTIES OF GREATEST INTEGER FUNCTION:

If n is an integer and x is any real number between n and n+1, then the greatest integer function has the following properties:

i) [-n] = - [n]

ii) [x+n] = [x] + n

iii) [-x] = -[x] - 1

iv) 

v) 

vi) [x] ≥ n => [x] ≥ n, where n ϵ Z

vii) [x] ≤ n => x < n+1, n ϵ Z

viii) [x] > n => x ≥ n+1, n ϵ Z

ix) [x] < n => x < n, n ϵ Z

x) [x+y] = [x] + [y + x – [x]] for all x, y ϵ R.

xi) 

SIGNUM FUNCTION:

The function defined by

Or, 

Is called the signum function.

The domain of the signum function is R and the range is the set {-1, 0, 1}. The graph of this function is shown in figure.

PERIODIC FUNCTIONS:

Periodic Functions:

A function f (x) is said to be a periodic function if there exists a positive real number T such that f (x + T) = f (x) for all x ϵ R.

We know that

sin (x + 2π) = sin (x + 4π) = … = sin x

And,

cos (x + 2π) = cos (x + 4π) = … = cos x for all x ϵ R.

Therefore,sinx and cosx are periodic functions.

PEROID:

If f (x) is a periodic function, then the smallest positive real number T is called the period or fundamental period of function f (x) if f (x + T) = f (x) for all x ϵ R..

In order to check the periodicity of a function f (x), we follow the following algorithm.

SOME USEFUL RESULTS ON PERIODIC FUNCTIONS:

RESULT 1:

If f (x) is a periodic function with periodic T and a, b ϵ R such that a ≠ 0, then a f (x) + b is periodic with period T.

RESULT 2:

If f (x) is a periodic function with period T and a, b ϵ R such that a ≠ 0, then f (ax + b) is periodic with period T / |a|.

RESULT 3:

Let f (x) and g (x) be two periodic functions such that:

Period of f(x) = m/n, where m, n ϵ N and m, n are co-prime.

And,

Period of g(x) = r/s, where r ϵ N and s ϵ N are co-prime.

Then, (f + g) (x) is periodic with period T given by T = LCM of (m, r) / HCF of (n, s)

Provided that there does not exist a positive number k < T for which f (k + x) = g(x) and g (k + x) = f(x) else k will be the period of (f + g) (x).

The above result is also true for functions f/g, f – g and fg.

Example:

Find the period of f (x) = 5sin3x - 7sin8x

Solution:

We observe that:

Period of 5sin 3x is 2π / 3

Period of 7 sin 8x is 2π / 8 = π/ 4

.·. Period of f (x) = LCM of 2π and π / HCF of 3 and 4= 2π / 1 = 2π