FEW IMPORTANT POINTS - MATHS

• sinⁿx, cosⁿx, secⁿx, cosecⁿx are periodic functions with period 2π and π according as n is odd or even.
• tanⁿx, cotⁿx are periodic functions with period π whether n is even or odd.
• |sin x|, |cos x|, |tan x|, |cot x|, |sec x|, |cosec x| are periodic with period π.
• |sin x| + |cos x|, |tan x| + |cot x|, |sec x| + |cosec x| are periodic with period π/2
• sin^-1 (sin x), cos^-1 (cos x), cosec^-1 (cosec x), sec^-1 (sec x) are periodic with period 2π whereas tan^-1 (tan x) and cot^-1 (cot x) are periodic with period π.

Example:

Find the period of the function f(x) = e ^{x - [x] + |cos πx| + |cos 2π x| + … + |cos n π x|}

Solution:

We observe that

Period of x – [x] is 1

Period of |cos π x| is π/π = 1

Period of |cos 2π x| is π/2π = 1/2  

Period of |cos 3π x| is π/3π = 1/3

And so on.

Finally, Period of |cos nπ x| = π/nπ = 1/n

.·. Period of f (x) = LCM of 

EVEN AND ODD FUNCTIONS

EVEN FUNCTIONS:

A function f (x) is said to be an even function if f (-x) = f (x) for all x.

ODD FUNCTIONS:

A function f (x) is said to be an odd function if f (-x) = - f (x) for all x.

Hence, the values of x are

(-1 + √5)/2, (-1 - √5) /2, (-3 - √5)/2 and (-3 + √5)/2.  

REMARK:

Let f, g be two functions. Then

1. f is even, g is even => fog is an even function.
2. f is odd, g  is odd => fog is an odd function.
3. f is even, g  is odd => fog is an even function
4. f is odd, g  is even => fog is an even function.