Determining if function odd or even 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


*

*what is f(-x)

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

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

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

*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?

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. 
