The Student Room Group
Uni students - Complete our survey >>
Uni students - Complete our survey >>
Uni students - Complete our survey >>

This discussion is now closed.

Check out other Related discussions

Function question

Hi,
Could anybody prove that for any function f(x)

f(x) - f(-x) is odd.

so far this is what i've done:

suppose f(x) is odd, then f(x) - f(-x) = -f(-x) - f(-x)=-2f(-x)=2f(x) which is odd

suppose f(x) is even, then f(x) - f(-x)= f(x) - f(x) = 0, which is supposed to be odd???

Thanks for any help
Reply 1
Avatar for Jonny W
Jonny W
8
Let g(x) = f(x) - f(-x).

Then

g(-x)
= f(-x) - f(-(-x))
= f(-x) - f(x)
= -[f(x) - f(-x)]
= -g(x)

So g is odd.
manhunter
Hi,
Could anybody prove that for any function f(x)

f(x) - f(-x) is odd.

so far this is what i've done:

suppose f(x) is odd, then f(x) - f(-x) = -f(-x) - f(-x)=-2f(-x)=2f(x) which is odd

suppose f(x) is even, then f(x) - f(-x)= f(x) - f(x) = 0, which is supposed to be odd???

Thanks for any help

let g(x)=f(x)-f(-x)
then g(-x)=f(-x)-f(x)=-(f(x)-f(-x))=-g(x)
Reply 3
Avatar for elpaw
elpaw
15
manhunter
suppose f(x) is even, then f(x) - f(-x)= f(x) - f(x) = 0, which is supposed to be odd???

It is odd. it is also even

what yoiu are assuming is that all functions are either even or odd.

follow the methods above.
Jonny W
Let g(x) = f(x) - f(-x).

Then

g(-x)
= f(-x) - f(-(-x))
= f(-x) - f(x)
= -[f(x) - f(-x)]
= -g(x)

So g is odd.



What makes it odd or even - how can you tell? :rolleyes: :confused: :biggrin:
Reply 5
Avatar for elpaw
elpaw
15
~U~Teedy~U~
What makes it odd or even - how can you tell? :rolleyes: :confused: :biggrin:

for an odd function: f(-x) == -f(x)

for an even function: f(-x) == f(x)
Reply 6
Avatar for El Stevo
El Stevo
11
well, you cannot state f(x) is even, or odd, but you can state that 2f(x) is even because 2 of anything is always even*

*if x=2p x implies x = even when p is any integer, (will always divide by 2), if x=p you dont know whether it is or not, and if x=2p-1 then x is odd because it defintely has a remaidner when divided by 2
Thanks guys!
Reply 8
Avatar for elpaw
elpaw
15
El Stevo
well, you cannot state f(x) is even, or odd, but you can state that 2f(x) is even because 2 of anything is always even*

but functions are continuous, not discrete (integers)

as a proof by counterexample, 2*sin(x) is not even.

Related discussions

Latest

Trending

Trending

Articles for you