Question

Show that fn(x) in even and odd function for even n and for odd n respectively.

Answer 1

First of all you must know the rule of even and odd functions:

A function f is said to be even or symmetric if f(x) = f(-x)

and asymmetric if f(x) = -f(-x).

As we know that a even number multiplied by any number is always even. for example : 2^3 = 2*2*2 = 8 and a odd number multiplied by a odd number is always odd example: 3^4 = 3*3*3*3 = 81

however, for all this to be true, the value of n must be greater than 1 ….. n > 1.

Answer 2

Answer:

The proof can be explained step wise below :

Step-by-step explanation:

We know any function f : A ---> B is given,

And if function , f(x) = -f(-x) then, f(x) is an odd function.

And if function, f(x) = f(-x) then, f(x) is an even function.

Now, we have to show , fⁿ(x) is even when n is even and fⁿ(x) is odd if n is odd.

Case 1 :- when n is even number.

then, fⁿ(x) = [f(x)]ⁿ

[f(-x)]ⁿ = [f(x)]ⁿ [If f(x) is even then, f(x) = f(-x) ]

Hence, fⁿ (x) is also an even function when n is even

Case 2 :- when n is odd number,

then, fⁿ(x) = [f(x)]ⁿ

[-f(-x)]ⁿ = -[f(-x)]ⁿ = -fⁿ(-x)

[ Because , an odd function f(x) = - f(-x)]

So, fⁿ(x) is an odd function when n is odd.