You should be able to solve this

Show me what ya got.

1) If f(x) + f\left(\frac{1}{1-x}\right) = x+1 for all x \ne 1, find a formula for f(x).

2) If a,b,c,d are distinct
numbers such that for some number k
a^4+a^2+ka+64=0
b^4+b^2+kb+64=0
c^4+c^2+kc+64=0
d^4+d^2+kd+64=0
find a^2 + b^2 + c^2 + d^2.

3) Let f be an infinitely differentiable function on [0,1).  Extend f to \mathbb{R^+} by defining f(x) = f(x-1) + f'(x-1) when x \ge 1.  Find a one term expression for f involving only elementary functions, the differentiation operator, the floor operator, and f evaluated on [0,1).

(inb4 the TeX's all fucked up)

>>11

Actually had an idea about it last night in bed (I need to get a life).

I'm pretty certain z, 1-z, 1/1-z, z/1-z, 1-z/z form a compositional group, or some small group like that, haven't used mobius maps in a while.

Anyway, presumably you can keep reapplying the definition to get

f(x) = g(x) - f(1/1-x)
f(x) = h(x) + f(x-1/x)
f(x) = i(x) - f(x)

as 1/1-(x-1/x) = x, actually turned out simpler than I thought.

Here g,h and i are rational functions in x that I can't be arsed to work out, but it's obvious where the answer will come from now.

This is one of the kind of threads that I read sci for. We should make the effort to grind out the other questions.