Let f and g be two odd functions. Prove that:

Question

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!

Answer 1

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.

Answer 2

got it.

Thank you so much!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!