Even and odd functions and whether even/odd characteristics change with powers

Question

I watched a video where a problem involved recognizing that $\sin x$ is an odd function and $\sin^3 x$ is also odd. But the presenter didn't explain why $\sin^3 x$ is also odd. Why does the fact that the function is odd not change when it is cubed? Is there a rule where for every even power the odd function is even and for every odd power the odd function remains odd? What about for even functions?

Answer 1

Note that $$(-1)^{2k}=(+1)$$ and $$(-1)^{2k+1}=(-1)$$ $$(+1)^k = (+1)$$

Thus if a function is odd we have $$f^{2k}(-x) = (-1)^{2k}f^{2k}(x)=f^{2k}(x)$$ and $$f^{2k+1}(-x) = (-1)^{2k+1}f ^{2k+1}(x)=-f^{2k+1}(x)$$

Thus odd functions to the odd powers are odd and to the even powers are even.

Even functions to any power stay even.

Answer 2

It is not only the cubing, it works for anyother odd function, it is always true that the composite of odd functions is an odd function.

In your case, $f(x)=\sin x$ is an odd function and $g(x)=x^3$ is also an odd function so $g\circ f(x)= g(f(x))=g(\sin x)=\sin^3x$

Another example is $\sin x$ with any odd power for example $x^{12345}$, since both are odd functions then $\sin^{12345}x$ is also an odd function.

Or $\frac{1}{\sin x}$ is also an odd function since it is the composite of $\sin x$ and $\frac{1}{x}$,and they are both odd functions.

Anyway you can just check if $f(-x)=-f(x)$ without worrying about compositions, $f(-x)=\sin^3(-x)=(\sin (-x))^3=(-\sin x)^3 =-(\sin^3x)=-f(x)$