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.

tomtom Jan 17, 2024

#1**+1 **

**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!**

Imcool Jan 18, 2024