Previous in Relation and Functions Next in Relation and Functions
Question Number 199167 by tri26112004 last updated on 28/Oct/23
![Give a function f: R→(0;+∞) continous on R and such that f(x+y) = f(x).f(y) a. Prove f(0) = 1 b. Let h(x) = ln[f(x)]. Prove that: h(x+y) = h(x) + h(y) c. Find all the function f such that problem request](Q199167.png)
$${Give}\:{a}\:{function}\: \\ $$$${f}:\:{R}\rightarrow\left(\mathrm{0};+\infty\right)\:{continous}\:{on}\:{R}\:{and}\:{such}\:{that} \\ $$$${f}\left({x}+{y}\right)\:=\:{f}\left({x}\right).{f}\left({y}\right) \\ $$$${a}.\:{Prove}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{1} \\ $$$${b}.\:{Let}\:{h}\left({x}\right)\:=\:{ln}\left[{f}\left({x}\right)\right].\:{Prove}\:{that}: \\ $$$$\:{h}\left({x}+{y}\right)\:=\:{h}\left({x}\right)\:+\:{h}\left({y}\right) \\ $$$${c}.\:{Find}\:{all}\:{the}\:{function}\:{f}\:{such}\:{that}\:{problem}\:{request} \\ $$$$\:\:\: \\ $$$$\: \\ $$