Understanding the Odd and Even Nature of sin(x^3)

[mech-eng]

TL;DR

Determination of a functon for being odd or even

Hello, would you please explain how to determine if sin $x^3$ is odd or even? Is there anyway to understand it without drawing the graph?

Thank you.

 

[phyzguy]

A function is odd if f(-x) = -f(x) and even if f(-x) = f(x). So what do you think of sin(x^3)? Or did you mean (sin(x))^3?

 

[mech-eng]

A function is odd if f(-x) = -f(x) and even if f(-x) = f(x). So what do you think of sin(x^3)? Or did you mean (sin(x))^3?

Thanks. I just could not have seen this so easily. Now I got it.

 

[PeroK]

Thanks. I just could not have seen this so easily. Now I got it.

It's interesting. If someone had asked me if a composition of two odd functions is odd or even I might have guessed even. But, if $f$ and $g$ are both odd, then:

$f(g(-x)) = f(-g(x)) = - f(g(x))$

Hence $f \circ g$ is odd. I guess it's like multiplying two odd numbers.

Also, what if we have an odd function and an even function. E.g. if $g$ is even and $f$ is odd:

$f(g(-x)) = f(g(x))$

Hence $f \circ g$ is even.

And, it's the same if you have any number of odd functions and one even function. A single even function kills all the oddness! The same as multiplication.

 

[TeethWhitener]

It's interesting. If someone had asked me if a composition of two odd functions is odd or even I might have guessed even. But, if $f$ and $g$ are both odd, then:

$f(g(-x)) = f(-g(x)) = - f(g(x))$

Hence $f \circ g$ is odd. I guess it's like multiplying two odd numbers.

Also, what if we have an odd function and an even function. E.g. if $g$ is even and $f$ is odd:

$f(g(-x)) = f(g(x))$

Hence $f \circ g$ is even.

And, it's the same if you have any number of odd functions and one even function. A single even function kills all the oddness! The same as multiplication.

$f\circ g$ is even regardless of whether $f$ is even, odd, or neither.

Edit: Question for someone who knows more about math than me: would the even functions be considered an ideal under composition?

 

[Mark44]

A function is odd if f(-x) = -f(x) and even if f(-x) = f(x).

I just could not have seen this so easily.

Why is that? phyzguy is just using the definitions of odd and even, the first things you should be looking for.

 

[Bosko]

-f(-x)
= -sin( (-x)^3)
= -sin( (-x) (-x) (-x) )
= -sin( - x^3) < - sin () is odd
= - ( - sin ( +x^3))
= sin ( x^3)
= f(x)
= +f(+x)

The function is odd.

 

[mech-eng]

Why is that? phyzguy is just using the definitions of odd and even, the first things you should be looking for.

Because I focused on graph or geometry to recognize them.

 

[Mark44]

Because I focused on graph or geometry to recognize them.

But you should also keep the definition in mind...