Posted by manny on Thursday, September 13, 2007 at 1:27pm.
Let f and g be two odd functions. Prove that:
a) f + g is an odd function
b) g of f is an odd function
I am not even sure where to start, any help that can be provided would be appreciated!

Math  Count Iblis, Thursday, September 13, 2007 at 1:38pm
A function f is odd iff
f(x) =  f(x)
Put h(x) = f(x) + g(x) and calculate
h(x):
h(x) = f(x) + g(x) =
f(x)  g(x) =
[f(x) + g(x)] =
h(x)
So, we see that h is odd because h(x) = h(x)
Now put h(x) = g[f(x)]
h(x) = g[f(x)] =
g[f(x)] =
g[f(x)] =
h(x)
And we see that h is odd.
One more exercise you could do:
If f(x) is an arbitrary function show that it can be decomposed uniquely as:
f(x) = f_even(x) + f_odd(x)
where f_even and f_odd are even and odd functions, respectively. Give the expressions for these functions in terms of the function f.

Math  manny, Thursday, September 13, 2007 at 1:45pm
got it.
Thank you so much!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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