Help.............

sorry akand i m replying so late

i ve my school phy exam tom too u see;0

u ve not specified domain so im taking it as R→R

k sub f(y)= t ,x=0

f(-t)=f(t)+f(0)-1 put t=0 , f(0)=1

now put f(y)=x in org eq

u get f(0)=f(x)+x²+f(x)-1

hence f(x)= 1 - x²/2

good try celestine but that is not the correct solution ...
first of all when u put t=0 then u mean to say that

f(y)=0 for some y ;
but 0 may not be in the image of A ...

then further

you put f(y)=x

hence you have found the f(x) for those x only who are in the image of the given function

this isnt such an easy problem ... ive encountered it before ..

it is from the international mathematics olympiad ...

dont forget we have to find all such functions ..

yes i knew that already
as akand dint specify any cond i solved it directly