Question

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

Answer 1

We know any function $f:A\rightarrow B$ is given,
And 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.