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

 Feb 28, 2024
 #1
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we need:
\(f(f(x))=x\\ f(x)=f^{-1}(x)\)
The only way this is possible is if \(f(x)=f^{-1}(x)=x\):
So:
\(\cfrac{cx}{(4x-5)(3x+2)}=x\\ \boxed{\displaystyle{c=(4x-5)(3x+2)}}\)

 Feb 28, 2024

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