Question

Match the following functions satisfying particular functional relationship

	Column-I
(A)	$f(x+y)=f\left(x\right)+f\left(y\right)$
(B)	$f\left(xy\right)=f\left(x\right)+f\left(y\right)$
(C)	$f(x+y)=f\left(x\right)\cdot f\left(y\right)$
(D)	$f\left(X\right)+f\left(y\right)=f\left(\frac{x+y}{1-xy}\right)$

	Column-II
(p)	${\log }_{3}x$
(q)	$ta{n}^{-1}x$
(r)	$3x$
(s)	$3^x$
(t)	A transcendental function

Answer 1

(A) $3(x+y)=3x+3y\Rightarrow f(x+y)=f\left(x\right)+f\left(y\right)$
(B) ${\log }_{3}xy={\log }_{3}x+{\log }_{3}y\Rightarrow f\left(xy\right)=f\left(x\right)+f\left(y\right)$
(C) $f(x+y)=f\left(x\right)\cdot f\left(y\right)\Rightarrow 3^{x+y}=3^x+3^y$
(D) ${\tan }^{-1}x+{\tan }^{-1}y={\tan }^{-1}\left(\frac{x+y}{1-xy}\right)$
$\Rightarrow f\left(x\right)+f\left(y\right)=f\left(\frac{x+y}{1-xy}\right)$