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The function
f(x) = \frac{cx}{4x - 5 - 2x + 8}
satisfies f(f(x))=x for all real numbers x\neq -\frac{3}{2}. Find c.

 Jan 17, 2024
 #1
avatar+266 
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Please take the time to put your answer in Latex. It helps a lot.

 

\(f(x) = \frac{cx}{4x - 5 - 2x + 8}\)

\( x\neq -\frac{3}{2}\)

 

Im going to try and brute force it, but its going to be really ugly.

\(f(f(x)) = \frac{c(\frac{cx}{4x - 5 - 2x + 8})}{4(\frac{cx}{4x - 5 - 2x + 8}) - 5 - 2(\frac{cx}{4x - 5 - 2x + 8}) + 8}\)

\(f(f(x)) = \frac{c(\frac{cx}{4x - 5 - 2x + 8})}{2(\frac{cx}{4x - 5 - 2x + 8}) +3}\)

\(x = {\frac{c^2x}{2x+3}}\cdot{\frac{2x+3}{2cx+6x+9} }\)

\(x = {\frac{c^2x}{2cx+6x+9} }\)

I think you can take it from here. Good Luck!

 Jan 18, 2024

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