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

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