Question

PLEASE HELP ME !!!!! Why is g(x)=x^3/sin(x) an odd function ?

Answer 1

Well, honestly I'm confused.
http://en.wikipedia.org/wiki/Even_and_odd_functions
By the explanation in here, an odd function is a function so:
\[
f(-x) = -f(x)
\]
for every X in the function's domain.

let's try:
\[
g(x) = \frac{x^3}{sin(x)}\\
g(1) = \frac{1^3}{sin(1)} = \frac{1}{sin(1)}\\
g(-1) = \frac{(-1)^3}{sin(-1)} = \frac{-1}{-sin(1)} = \frac{1}{sin(1)}
\]
And...
\[
\frac{1}{sin(1)} \ne -\bigg[ \frac{1}{sin(1)} \bigg]
\]
So..
\[
g(-1) \ne -g(1)
\]
And therefore it can't be odd function by the definition... it's actually pretty easy to see that it is even function as
\[
g(x) = g(-x)
\]
in here..

They also write there:
"Examples of odd functions are x, x3, sin(x), sinh(x), and erf(x)."
And
"The quotient of two odd functions is an even function."

Answer 2

but if you graph the eqution it looks like it is on the origin which is an odd function

Answer 3

You mean it goes through (0,0)?

Answer 4

I don't see it that way O.o... sin(0) should be... 0.... and we can't divide by 0. maybe it should be cos(x) instead?

Answer 5

well if you do the odd function test g(-x)

Answer 6

and the -g(x)

Answer 7

like i did above? besides \( x \ne 0 \)
and eh...
http://www.wolframalpha.com/input/?i=x%5E3%2Fsinx
They even say there its parity is 'even'.. idk what else you mean then

Answer 8

Oh ! yeah thank you !

Answer 9

I just wanted to see how to come about the answer algebraically

Answer 10

so what would be f(x)= 0.001x^5+10x-57 ?

Answer 11

Ok.
\[
x^3 = x \cdot x \cdot x \\
(-x)^3 = (-x) \cdot (-x) \cdot (-x) = -(x \cdot x \cdot x) \\
\text{Therefore:} \\
-(x^3) = (-x)^3 \quad \quad \to \quad\quad-(x \cdot x \cdot x) = -(x \cdot x \cdot x)
\]
Now... it's known that, and easy to see also that...
\[
sin(-x) = -sin(x)
\]
Therefore they both are odd functions... means if \(x < 0 \) they would produce negative value..
and negative/negative is positive...
so
\[
g(-x) = \frac{(-x)^3}{sin(-x)} = \frac{x^3}{sin(x)} = g(x) \\
\text{So:} \\
g(-x) = g(x)
\]
Means... g(x) is even.

Answer 12

I see ! so would this equation be neither f(x)=0.001x^5+10x-57

Answer 13

\[
f(x) = 0.001x^5+10x-57
\]
well.. I guess it would be neither.. since
\[
f(1) = 0.001+ 10 - 57 = 0.001 - 47 = -46.999 \\
f(-1) = -0.001 - 10 - 57 = -(0.001 + 67) = -67.001\\
-(-46.999) \ne -67.001\\
46.999 \ne -67.001\\
-f(1) \ne f(-1)
\]
Besides...
http://www.wolframalpha.com/input/?i=+0.001x%5E5%2B10x-57
No parity mentioned

Answer 14

I should also add for even, very sorry
\[
-46.999 \ne -67.001 \\
f(1) \ne f(-1)
\]
There for this is not even and not odd(above).

Answer 15

Okay thank you so much !!!

Answer 16

Anything else?

Answer 17

Nope :) thanks I think I understand now

Answer 18

Sure np, good luck then