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# If $f$ is a real values function satisfying $f(x+y)=f(x)\;f(y)$ and $f(1)=a$ where $x,y\in R$, then $f(x)$ =

(A) $x^a$
(B) $a^{2x}$
(C) $a^{x/2}$
(D) $a^x$
Can you answer this question?

Ans- (D)
Given $f(1)=a$ and $f(x+y)=f(x)f(y)$
$\Rightarrow\:f(2)=f(1+1)=f(1)f(1)=a^2$
$\Rightarrow f(3)=f(2+1)=f(2)f(1)=a^3$
$\Rightarrow f(4)=a^4 \rightarrow f(x)=a^x$
answered May 4, 2013
edited May 17, 2014