Proving the product of three odd funtions is odd.

[sirhc]

Hey,

Just looking for some help on the below question, I have absolutely no idea.

"Prove that the product of three odd functions is odd."

Thanks in advance ,
sirhc

 

[Deleted member 4993]

Hey,

Just looking for some help on the below question, I have absolutely no idea.

"Prove that the product of three odd functions is odd."

Thanks in advance ,
sirhc

Hint: A general odd function is expressed as (2n+1)

 

[ksdhart2]

Hint: A general odd function is expressed as (2n+1)

Er... hold on. It's true that a generalized odd number can be written as (2n+1), but the question asks about odd functions. To that end, since the OP claims to have "absolutely no idea," a good place to start is by reviewing the definitions. Two webpages I've found that cover this topic in a nice way are this page from the The University of Tennessee Knoxville's math department, and this page from PurpleMathhttp://www.purplemath.com/modules/fcnnot3.htm. UTK's webpage notes that:

DEFINITION.
A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.
A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.

Woflram MathWorldhttp://mathworld.wolfram.com/OddFunction.html adds that "Since an odd function is zero at the origin, it follows that the Maclaurin serieshttp://mathworld.wolfram.com/MaclaurinSeries.html of an odd function contains only odd powers." and this page from Purduehttp://www.math.purdue.edu/~krotz/teaching/evenoddfourier.pdf notes that:

Here are some useful properties of even and odd functions. All of them are easy to check from the definition of even and odd functions.

1. A product of two even functions is even. Therefore, the function x^2 * cos(2x) is even since both of its factors are even.
2. A product of two odd functions is even. Thus x^3 * sin x is even.
3. A product of an odd function with an even function is odd. So the function x cos(x) is odd since x is odd and cos(x) is even.

As a warm-up exercise, maybe try proving the above three statements, then see where all of the information you've accumulated thus far leads you. If you get stuck again, that's alright, but when you reply back, please include any and all work you've done on this problem, even the parts you know for sure are wrong. Thank you.

 

[siavash533]

Proving the product of three odd funtions

Hey,

Just looking for some help on the below question, I have absolutely no idea.

"Prove that the product of three odd functions is odd."

Thanks in advance ,
sirhc

In an odd function z(x), z(-x)=-z(x)
If f(x) ,g(x), h(x) are three odd function we calculate product of three function as z(x)
z(x)= f(x).g(x).h(x) now calculate z(-x) hence:
z(-x)=f(-x).g(-x).h(-x) since f(x),g(x) and h(x) are odd function therefore we have
f(-x)=-f(x), g(-x)=-g(x) and h(-x)= -h(x) and substitute in z(-x) function
z(-x)=-f(x).g(x).h(x) therefore z(-x)=-z(x) therefore the product of three odd functions is an odd function.