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

I don't really understand this hoping if someone can help clarify this to me and how to do it so I know how to do the next problem!

 Jul 7, 2022
 #1
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Since f(x) = (ax + b)/(cx + d), the equation f(f(x)) = x becomes

(a*(ax + b)/(cx + d) + b)/(c*(ax + b)/(cx + d) + d) = x.

 

Cross-multiplying and simplifying, we end up with

a^2*x + ab - acx^2 - ax + bd - b - cdx^2 + cx^2 - d^2 x + dx = 0.

 

This factors as (a + d - 1)(-cx^2 + ax - dx + b) = 0.

 

We can't have -cx^2 + ax - dx + b constantly 0, so a + d - 1 = 0.  Therefore, a + d = 1.

 Jul 7, 2022
 #2
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I understood what you did but when i put the answer in it said it was incorrect so I'm not sure if its right?

Guest Jul 7, 2022

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