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If f(x) = (ax + b)/(cx + d), abcd =/= 0, and f(f(x)) = x for all x in the domain of f, what is the value of a + d?

 

Thank you!

 Sep 29, 2019
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\(f(f(x)) = x\\ f(x) = f^{-1}(x)\\~\\ y = \dfrac{ax+b}{cx+d}\\ cxy+dy = ax + b\\ (cy-a)x = b-dy\\ x = \dfrac{b-dy}{cy-a}\\ f^{-1}(x) = \dfrac{b-dx}{cx-a}\)

 

\(\dfrac{ax+b}{cx+d} = \dfrac{b-dx}{cx-a}\\ acx^2 +(bc-a^2)x-ab = -cd x^2 +(b c - d^2)x + bd\\ ac = -cd\\ bc-a^2 = bc - d^2\\ -ab = bd\\ a = -d\\~\\ a+d = 0\)

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 Sep 30, 2019

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