Mathematics - Functions Question with Solution | TestHub

The given functional equation is $f(x+y)=f(x)f(y)$. To find $f(0)$, let $x=0,y=0$. Then $f(0+0)=f(0)f(0)⟹f(0)=(f(0){)}^2$. This means $f(0)=0$ or $f(0)=1$. If $f(0)=0$, then for any $x$, $f(x)=f(x+0)=f(x)f(0)=f(x)\cdot 0=0$. This would imply $f(1)=0$, which contradicts $f(1)=3$. Therefore, $f(0)$ must be $1$. Now, to find $f(2)$, we can use $f(2)=f(1+1)=f(1)f(1)=3\cdot 3=9$. So, $f(0)+f(2)=1+9=10$. Thus, option A is correct.