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

In summary, when determining if a function is odd or even, we can use the definition that an odd function satisfies f(-x) = -f(x) and an even function satisfies f(-x) = f(x). In the case of sin(x^3), it is odd because -sin(-x^3) = sin(x^3). This can also be understood without graphing by looking at the composition of two odd functions, which results in an odd function, and the composition of an odd and even function, which results in an even function. An even function always dominates over an odd function when composing multiple functions.
  • #1
mech-eng
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TL;DR 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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  • #2
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
phyzguy said:
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
mech-eng said:
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
PeroK said:
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
phyzguy said:
A function is odd if f(-x) = -f(x) and even if f(-x) = f(x).

mech-eng said:
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
-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
Mark44 said:
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
mech-eng said:
Because I focused on graph or geometry to recognize them.
But you should also keep the definition in mind...
 
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1. What is the definition of an odd function?

An odd function is a mathematical function that satisfies the property f(-x) = -f(x), meaning that when the input is multiplied by -1, the output is also multiplied by -1. This results in the function being symmetric about the origin (0,0) and having rotational symmetry of 180 degrees.

2. What is the definition of an even function?

An even function is a mathematical function that satisfies the property f(-x) = f(x), meaning that when the input is multiplied by -1, the output remains the same. This results in the function being symmetric about the y-axis and having rotational symmetry of 180 degrees.

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

Sin(##x^3##) is an odd function. This can be seen by substituting -x for x in the function, resulting in sin((-x)^3) = sin(-x^3) = -sin(x^3). This satisfies the definition of an odd function, as the output is multiplied by -1.

4. How can I determine if a function is odd or even?

To determine if a function is odd or even, you can substitute -x for x in the function and see if the resulting function is equal to the original function multiplied by -1 (odd) or remains the same (even). Alternatively, you can graph the function and see if it is symmetric about the origin (odd) or y-axis (even).

5. What are some examples of odd and even functions?

Examples of odd functions include sin(x), cos(x), and x^3. Examples of even functions include x^2, cosh(x), and e^x. Note that not all functions are strictly odd or even, and some may have both odd and even components.

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