If f,is an odd real function and g an even real function, then their composition is an odd real function

Prove that:
a) If f,is an odd real function and g an even real function, then their composition is an odd real function

Proof
f(-x)=-f(x) , for every x in R
g(-x)=g(x),for every x in R

(f*g)(-x)=f(g(-x))=f((-g(x))=-f(g(x))=-(f*g)(x)