Question

If $f(x+y)=f\left(x\right).f\left(y\right)$ then prove that $f\left(x\right)={\left[f\left(1\right)\right]}^x$

Answer 1

$f(x+y)=f\left(x\right).f\left(y\right)f\left(2\right)=f\left(1\right).f\left(1\right)=\left[f\right(1\left){]}^{2}f\right(3)=f(2).f(1)=[f\left(1\right){]}^3\Rightarrow f\left(x\right)=\left[f\right(1){]}^x$