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This exercise on the Khan Academy requires you to determine whether the following function is odd or even

f(x) = $-5x^5 - 2x - 2x^3$

To answer the question, the instructor goes through the following process

  1. what is f(-x)
  2. f(-x) = $-5(-x)^5 - 2(-x) - 2(-x)^3$
  3. f(-x) = $5(x)^5 +2(x) + 2(x)^3$
  4. f(-x) = $-(-5x^5 - 2x - 2x^3)$
  5. f(-x) = -f(x)

Therefore f(x) is odd.

Question. Why is everything converted to a negative both inside and outside the parenthesis at step 4?

At step 2, he's making x negative. At step 3, we get the result of making the x values negative. So why add the negatives again at step 4?

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  • $\begingroup$ Note that having a minus sign does not make $-x$ negative. For example if $x=-1$, we have $-x=1$ which is perfectly positive. $\endgroup$ – Taladris Aug 4 '15 at 15:03
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    $\begingroup$ @MichaelGaluza it is not a typo. In step 4, a double negation is applied. $\endgroup$ – Zoltán Aug 4 '15 at 16:08
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If you have troubles with this approach, what you can do instead is the following:

Step 1 Calculate $f(-x)$. This is 1.-3. in your computation.

Step 2. Calculate $-f(x)$. In this case

$$-f(x)=-(-5x^5 - 2x - 2x^3)=5x^5+2x+2x^3$$

Step 3 Compare. If they are equal, the function is odd, if not, it is not odd.

[ Step 4 Compare $f(-x)$ to $f(x)$. If they are equal, the function is even, if not, not.]

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Let's follow the simplest process

Notice, $$f(x)=-5x^5-2x-2x^3$$ $$f(-x)=-5(-x)^5-2(-x)-2(-x)^3$$ $$ =5x^5+2x+2x^3$$ $$\implies f(x)+f(-x)=-5x^5-2x-2x^3+5x^5+2x+2x^3=0$$ Hence, $f(x)$ is odd.

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  • $\begingroup$ this is helpful but I'm also looking for a simple explanation (in words hopefully) as to why all the negatives are added inside and outside the parenthesis at step 4. Hint, I'm not very good at math. $\endgroup$ – Michael Aug 4 '15 at 13:21
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we have $$f(-x)=-5(-x)^5-2(-x)-2(-x)^3=5x^5+2x+2x^3=-f(x)$$ since $$(-x)^5=-x^5$$ $$-x=-x$$ and $$(-x)^3=-x^3$$

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    $\begingroup$ it shows the derivation again..but it is not providing anymore insight to the OP. $\endgroup$ – Chinny84 Aug 4 '15 at 13:16
  • $\begingroup$ the other answer also $\endgroup$ – Dr. Sonnhard Graubner Aug 4 '15 at 13:18
  • $\begingroup$ I also agree. But unfortunately(fortunately depending on the changes) you were first. I will also state the same on his. $\endgroup$ – Chinny84 Aug 4 '15 at 13:19
  • $\begingroup$ thank you but I'm looking for a simple explanation (in words hopefully) as to why all the negatives are added inside and outside the parenthesis at step 4. $\endgroup$ – Michael Aug 4 '15 at 13:21
  • $\begingroup$ both answers give a correct way to solve the problem from above $\endgroup$ – Dr. Sonnhard Graubner Aug 4 '15 at 13:21
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In step 4, the instructor is factoring out a $-1$.

Just like $2a + 4b + 6c$ can be factored into $2(a + 2b + 3c)$, so can $5x^5 +2x + 2x^3$ be factored into $-1(-5x^5 - 2x - 2x^3)$. He just omitted the $1$ when writing it.

The reason he factored out a $-1$ is because that makes the inner expression the same as $f(x)$, and he's attempting to prove that $f(-x) = -f(x)$.

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