Question

If f:R--> R be a function given by

f(x+y) = f(x) f(y) for all x,y belongs to R.
Prove that f(x) = [f(1)]^x for all x belongs to R.

Answer 1

Dear Student

$$\begin{gathered} \mathrm{Given} \\ \mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}\left(\mathrm{x}\right)\mathrm{f}\left(\mathrm{y}\right) \\ \mathrm{This}\mathrm{functional}\mathrm{equation}\mathrm{is}\mathrm{satisfied}\mathrm{by}\mathrm{the}\mathrm{function} \\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{a}}^{\mathrm{x}} \\ \mathrm{Therefore}\mathrm{f}\left(1\right)=\mathrm{a} \\ \mathrm{Replace}\mathrm{a}\mathrm{with}\mathrm{f}\left(\mathrm{x}\right)\mathrm{in}\mathrm{original}\mathrm{equation},\mathrm{you}\mathrm{get} \\ \mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(1{)}^{\mathrm{x}} \\ \mathrm{Hence}\mathrm{proved} \end{gathered}$$
Regards