1. Tardigrade
  2. Question
  3. Mathematics
  4. For an arbitrary function f with domain (-∞, ∞), define F ( x )= f ( x )+ f (- x ) and G ( x )= f ( x )- f (- x ). Which of the following MUST be an odd function?

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Q. 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?

Relations and Functions - Part 2

Solution:

(A) $(F+G)(x)=F(x)+G(x)=2 f(x)$, (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.