Question

What is functional equation of \[exp(x+y)\] and \[x,y \in \mathbb{R}\] and conclude from solution \[\sqrt{exp(x)} = exp(\frac{x}{2})\]

Answer 1

i think functional equation from \[exp(x+y)\] is \[exp(x+y)=exp(x).exp(y)\] but i dont know how to conclude from that \[\sqrt{exp(x)}=exp(\frac{x}{2})\]

Answer 2

isnt a funciton equation something like:

f(x,y,...,k) = 0

Answer 3

we know that:\[\sqrt{e^x}=(e^x)^{1/2}=e^{x/2}\]

Answer 4

Nice explaining :)

Answer 5

hmm maybe, i am not sure but i can write what in my german book is as definition for functional equation of exponential function is like that \[e^{x+y}={e^{x}}.{e^{y}} \forall x, y \in \mathbb{R} \]

Answer 6

wikipedia is saying your f(x+y) = f(x)f(y) definition is fine as well

Answer 7

ooh ok, also what would be the functional equation of \[ \exp(x+y)\] ?

Answer 8

hmm, i dont think i have a good idea on how to process this just yet.

exp(x+y) = exp(x) exp(y)

if we let x=y then

exp(x+x) = exp(x) exp(x)

exp(2x) = exp^2(x)

is what im thinking of

Answer 9

ok amistre , thank you very much for nice explanation

Answer 10

good luck :)