Odd / Even Functions

• May 17th 2011, 11:41 AM
bugatti79
Odd / Even Functions
Folks,

How does one show that f(x)sin(x) is an even function?

I attempt it be replacing x by -x ie

f(-x)sin(-x) but sin(-x) =-sin(x) therefore

f(-x)(-sin(x))=-f(-x)sin(x)????? (Dull)

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For the odd function g(x)cos(x) I calculate by replacing x by -x ie

g(-x)cos(-x) but cos(-x) = cos (x) therefore

g(-x)cos(x)=-g(x)cos(x) hence odd because g(-x)=-g(x)
• May 17th 2011, 11:47 AM
SpringFan25
Quote:

How does one show that f(x)sin(x) is an even function?
you forgot to tell us whether or not f(x) is an even function. I assume it is odd:

f(-x)sin(-x) =-1*f(x)*-1*sin(x) = f(x)sin(x)

Quote:

For the odd function g(x)cos(x) I calculate by replacing x by -x ie

g(-x)cos(-x) but cos(-x) = cos (x) therefore

g(-x)cos(x)=-g(x)cos(x) hence odd because g(-x)=-g(x)
agree.
• May 17th 2011, 11:53 AM
bugatti79
Quote:

Originally Posted by SpringFan25
you forgot to tell us whether or not f(x) is an even function. I assume it is odd:

f(-x)sin(-x) =-1*f(x)*-1*sin(x) = f(x)sin(x)

agree.

No f(x) is even...sorry for lack of clarity!
• May 17th 2011, 11:57 AM
SpringFan25
the product of an even function and an odd function is an odd function.

f(-x)sin(-x) = f(x)*-1*sin(x) = -f(x)sin(x).

so you wont be able to show the product is even.
• May 17th 2011, 12:00 PM
Plato
Quote:

Originally Posted by bugatti79
No f(x) is even...sorry for lack of clarity!

Then it is not true.
$\text{If }f(x)\text{ is even then }f(x)\sin(x)\text{ is odd.}$