Is sin(##x^3##) odd or even?

  • #1
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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.
 
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Answers and Replies

  • #2
phyzguy
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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?
 
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  • #3
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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.
 
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  • #4
PeroK
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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.
 
  • #5
TeethWhitener
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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?
 
  • #6
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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.
 
  • #7
Bosko
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-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.
 
  • #8
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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.
 
  • #9
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Because I focused on graph or geometry to recognize them.
But you should also keep the definition in mind...
 
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