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 \(A + B\)?
+23246 f(x) = Ax + B g(x) = Bx +A A <> B
f( g(x) ) = f( Bx + A) = A(Bx + A) + B = ABx + A2 + B
g( f(x) ) = g( Ax + B ) = B(Ax + B) + A = ABx + B2 + A
f( g(s) ) - g( f(x) ) = [ ABx + A2 + B ] - [ ABx + B2 + A ] = ABx + A2 + B - ABx - B2 - A = A2 - B2 + B - A
Since f( g(s) ) - g( f(x) ) = B - A
A2 - B2 + B - A = B - A
A2 - B2 = 0
(A + B)(A - B) = 0
Either A + B = 0
or A - B = 0 ---> A = B (but this is impossible because the problem states that A <> B)
So: A + B = 0
