While solving olympiad functional equation sometime we guess the answer and to verify it we put it back in the equation.
My question is say we want $F(X)$ and I have guessed $f(x)$ as answere now in books sometime they put $F(X)=f(x)+g(x)$ and then solve for $g(x)$ and sometimes they put $F(X)=f(x)g(x)$ and then solve for $g(x)$ , I want to know how to decide what to put back in the equation.
If required I can give examples:
Here is one https://artofproblemsolving.com/community/c6h1804289p11993777 and https://mail.google.com/mail/u/0?ui=2&ik=650d20114d&attid=0.1&permmsgid=msg-f:1653993712802343065&th=16f42a84c0502899&view=att&disp=safe&realattid=16f42a7d08264c9ee8f1
also why is there at all need to take g(x) , i mean once we have guessed a function we can just put the function as it is in the equation to verify, why involve g(x) at all