**EVEN AND ODD FUNCTIONS**

If f(x)=f(-x), then f(x) is even function

and if f(x) =-f(-x) then f(x) is odd function

For example Sin(x) is odd function while cos(x) is even function. This means that sin(x)=-sin(-x) and cos(x)=cos(-x)

With this basic understanding, let us see this question.

**Problem**: If function f satisfies the relation f(x+1)+f(1-x)=2f(x) for all real values of x and f(0)< or >0, then prove that f(x) is even.

**Solution:**

f(x+1)+f(1-x)=2f(x)

Replacing x with -x

f(1-x)+f(1+x)=2f(-x)

Comparing both we get, 2 f(x) = 2 f(-x)

Hence f(x) = f(-x)

Hence proved that, f(x) is an even function.

**COMPOSITE FUNCTIONS**

Let f : A à B, and g : B à C, be two functions, then gof : A à C

This is known as the product function or composite of f and g, given by gof(x)=g{f(x)} for all real values of x.

The symbols f : A à B is read as Function F maps A to B.

It is to be noted that fog = f{g (x)} and (f ± g) x = f(x) ± g(x) and (f/g)x=f(x)/g(x)

Example:

If F : RàR and G : RàR, be two mappings such that f(x) = sin x and g(x)=x

^{2}Then show that fog ≠ gof

Solution : Let x € R,

So (fog) x = f {g(x)} = sin(x

^{2}) -----(1)While (gof) x = g {f(x)} = g {sinx} = (sinx)

^{2 }= sin^{2}x -----(2)From (1) & (2), (fog) x ≠ (gof) x

**PERIODIC FUNCTIONS**

A function f(x) is said to be periodic function of x, if there exists a positive real number T such that f(x+T) = f(x). Then the smallest value of T is called the Period of the function.

For example Sin x is periodic because

Sin(2π+x)= Sin(x) and also sin(4π+x)=Sin(x) = Sin(6π+x) and so on…. But 2π being the smallest value is the period of the function.

Example: Show tha cos(√x) is non-periodic.

Solution: Let Cos(√x) be periodic with T as the period.

So, Cos(√x) = Cos{ √ (T+x) }

à √ (T+x) = 2nπ ± √x

Putting x = 0, we get, √T= 2nπ -----(1)

Putting x=T, we get √2T = 2nπ ± √T ----(2)

From (1) and (2) we get, √2T = √T ± √T

Or √T x √2 = √T (1 ± 1)

Or √2 = 1 ± 1, which is not possible, Hence Cos√x is not periodic function.

IIT JEE EXAM 2011- EXAM DATE 10 April 2011, Last date to Apply = 15 December 2010

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