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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. 

 Feb 12, 2019
 #1
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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

 

 

cool cool cool

 Feb 12, 2019

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