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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 - ,

    A function f is odd iff

    f(-x) = - f(x)

    Put h(x) = f(x) + g(x) and calculate

    h(-x) = f(-x) + g(-x) =

    -f(x) - g(x) =

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


    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 - ,

    got it.

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

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