Question

For an arbitrary function $f$ with domain $(-\infty ,\infty )$, define $F(x)=f(x)+f(-x)$ and $G(x)=f(x)-f(-x)$. Which of the following MUST be an odd function?

• A. $F+G$
• B. FG
• C. $\frac{F}{G}$
• D. G O G

Answer 1

Answer: (B), (C), (D)

(A) $(F+G)(x)=F(x)+G(x)=2f(x)$, (nothing definite can be said) [Special Test-3]
(B) (FG) (x) $=F(x)G(x)=[f(x)+f(-x)][f(x)-f(-x)]=[f(x){]}^2-[f(-x){]}^2$, which is clearly odd.
(C) $\left(\frac{F}{G}\right)(x)=\frac{F(x)}{G(x)}=\frac{f(x)+f(-x)}{f(x)-f(-x)}$, which is also clearly odd.
(D) $(GOG)(x)=G(G(x))=G(f(x)-f(-x))=f(f(x)-f(-x))-f(f(-x)-f(x))$, which is clearly odd.