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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 $A + B$?

 May 3, 2018
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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 $A + B$?

 

\(\begin{array}{|lrcll|} \hline f(x) = Ax + B \\ g(x) = Bx + A \\\\ f(g(x)) = f(Bx+A) = A(Bx+A)+B \\ g(f(x)) = g(Ax+B) = B(Ax+B)+A \\\\ f(g(x)) - g(f(x)) = A(Bx+A)+B - [B(Ax+B)+A] &=& B-A \\\\ \begin{array}{rcll} A(Bx+A)+B - [B(Ax+B)+A] &=& B-A \\ A(Bx+A)+B - B(Ax+B)-A &=& B-A \\ A(Bx+A) - B(Ax+B) &=& 0 \\ A(Bx+A) &=& B(Ax+B) \\ ABx+A^2 &=& BAx+B^2 \\ A^2 &=& B^2 \\ A^2-B^2 &=& 0\\ (A-B)(A+B) &=& 0 \\ A+B &=& \dfrac{0}{A-B} \quad | \quad A \neq B \ !\\ \mathbf{ A+B } & \mathbf{=} & \mathbf{0} \\ \end{array} \\ \hline \end{array} \)

 

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 May 3, 2018

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