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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?

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    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 06 '19 at 02:14
  • @amsmath Why are you allowed to put the the negative sign from inside the parentheses to outside? –  Oct 06 '19 at 02:16
  • Because $(ab)^n = a^n\cdot b^n$. – amsmath Oct 06 '19 at 02:16
  • 532874, that's the definition of odd function, no? – Gerry Myerson Oct 06 '19 at 02:17
  • @user532874 A function $f$ is odd if $f(-x) = -f(x)$ for all $x$. – amsmath Oct 06 '19 at 02:18
  • @amsmath Actually I still don't get $f(-x)^3=(-f(x))^3$. You are trying to prove that $g(-x)=-g(x)$ so you can't assume in the proof of $g(-x)=-g(x)$ that $g(-x)=-g(x)$. –  Oct 06 '19 at 02:31
  • @user532874 This is trivial. You have $f(-x) = -f(x)$. Now cube. – amsmath Oct 06 '19 at 02:35
  • @amsmath I'm getting confused with the notation. Let's work with the real function. How do I transform $\sin^3 (-x)$ into $-sin^3 (x)$? –  Oct 06 '19 at 02:44
  • Is it so hard to replace $f$ by $\sin$? – amsmath Oct 06 '19 at 02:45
  • @amsmath Oops I read $f(-x)^3=(-f(x))^3$ as $f(-x)^3=-f(x)^3$. $-f(x)^3$ is just $-g(x)$. So, I thought you were trying to prove $g(-x)=-g(x)$ by simply renaming the equation as $f(-x)^3=-f(x)^3$ which isn't a proof. –  Oct 06 '19 at 02:55
  • @user532874 No prob. – amsmath Oct 06 '19 at 02:57
  • @amsmath One more question. When you say $f^3$ I can't grasp the actual meaning of that. $f$ is a conceptual machine whose inputs are $x$ and whose outputs are $f(x)$. Thus, you can't cube a machine but you can cube its outputs. –  Oct 06 '19 at 03:23
  • @user532874 You're right. However, it is a convention to write $f^3$ for the machine that first outputs $f(x)$ and then cubes this value. – amsmath Oct 06 '19 at 03:25
  • @amsmath I feel that what actually constitutes a function is the most worst taught thing in school for math. I only realized a function is $f$ and not $f(x)$ 6 months ago and im a college freshman. Every problem always says "Let $f(x)$ be the function" but its not the function though. So messed up. –  Oct 06 '19 at 03:27
  • @user532874 I had the same problem in school. I always wondered what this mysterious $f(x)$ was that I couldn't understand. You're right -- the function is $f$, not $f(x)$. If you have some $x$, then $f(x)$ is the output value. I think it is didactically very bad what they do in school. – amsmath Oct 06 '19 at 03:31
  • @amsmath agreed. –  Oct 06 '19 at 03:35

2 Answers2

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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.

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

Fareed Abi Farraj
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