Prove that (f+g)(x) is an odd function, if f and g are odd functions (Stewart, Calculus)
Suppose f(x) and g(x) are odd functions. Prove that (f+g)(x) is also an odd function. Answer: 1. Strategy By definition, f is an odd function if and only if f(-x) = - f(x) To show (f+g) is an odd function, we need to show (f+g)(-x) = - (f+g)(x) 2. Explanation Since f(x) and g(x) are odd functions ⇒f(−x)=−f(x) and g(−x)=−g(x) By definition of sum of functions. (f+g)(−x)=f(−x)+g(−x) =−f(x)−g(x) =−(f(x)+g(x)) =−(f+g)(x) (by definition of sum of functions) ⇒(f+g)(−x)=−(f+g)(x) ⇒f+g is an odd function. Q.E.D.