nietzsche
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Homework Statement
For which numbers a, b, c, and d will the function
<br /> f(x) &= \frac{ax+b}{cx+d}<br />
satisfy f(f(x)) = x for all x (for which this equation makes sense.
Homework Equations
The Attempt at a Solution
I'm not sure if the answer will consist of specific values, or just a general solution, but here is my attempt so far:
<br /> \begin{align*}<br /> f(f(x)) &= \frac{a\frac{ax+b}{cx+d} + b}{c\frac{ax+b}{cx+d} + d}\\<br /> x &= \frac{a^2x + ab + bcx + bd}{cax + cb + dcx + d^2}\\<br /> 0 &= cx^2 + (d-a)x - b<br /> \end{align*}<br />
I skipped a bunch of steps in between, but that was the final equation I ended up with after simplification. Which was really weird, because I don't know what to do with that. If I had to guess, from the last equation, a, b, c, d must be values that satisfy the equation
<br /> x = \frac{ax+b}{cx+d}<br />
but again, I'm really confused and am not sure how to proceed.