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

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?

• If $f$ is odd and $g = f^3$, then $g(-x) = f(-x)^3 = (-f(x))^3 = (-1)^3f(x)^3 = -f(x)^3 = -g(x)$. You can play that game now with all variants. – amsmath Oct 6 '19 at 2:14
• @amsmath Why are you allowed to put the the negative sign from inside the parentheses to outside? – user532874 Oct 6 '19 at 2:16
• Because $(ab)^n = a^n\cdot b^n$. – amsmath Oct 6 '19 at 2:16
• 532874, that's the definition of odd function, no? – Gerry Myerson Oct 6 '19 at 2:17
• @user532874 A function $f$ is odd if $f(-x) = -f(x)$ for all $x$. – amsmath Oct 6 '19 at 2:18

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.

• I think you are missing a $2k$ exponent after "Thus if a function is odd we have" – user532874 Oct 6 '19 at 2:35
• @user532874 is it fixed now? – Mohammad Riazi-Kermani Oct 6 '19 at 2:43
• @MohammadRiazi-Kermani Check your calculations. You are missing minuses. – amsmath Oct 6 '19 at 2:59
• @amsmath Thanks for informative comment. I should have been more careful. – Mohammad Riazi-Kermani Oct 6 '19 at 3:12

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, $$f(-x)=\sin^3(-x)=(\sin (-x))^3=(-\sin x)^3 =-(\sin^3x)=-f(x)$$