Let $f(x) = Ax + B$ and $g(x) = Bx + A$, where $A \neq B$. If $f(g(x)) - g(f(x)) = B - A$, what is $A + B$?
ignore the dollar signs.
f(g(x) ) = A(Bx + A) + B
g(f(x)) = B(Ax + B) + A
So
f(g(x)) - g (f(x)) = B - A
[ A(Bx + A) + B ] - [ B(Ax + B) + A ] = B - A
ABx + A^2 + B - ABx - B^2 - A = B - A
A^2 - B^2 + B - A = B - A
A^2 - B^2 = 0 factor
(A + B ) ( A - B) = 0
This implies that A + B = 0
