B Is sin($x^3$) odd or even?

mech-eng

Summary
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.

Last edited:

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

Homework Helper
Gold Member
2018 Award
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

Gold Member
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

Mentor
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

Gold Member
-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

Mentor
Because I focused on graph or geometry to recognize them.
But you should also keep the definition in mind...

"Is sin($x^3$) odd or even?"

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