Math Circles on Fractals

Winter 2005

University of Waterloo - Waterloo, Ontario, Canada

Homework 3

1. Find the fixed points of each of the following real functions.
   (a) f(x) = 4x + 2
   (b) f(x) = (x + 1)/2
   (c) f(x) = x² - (3/4)
2. What happens to iterates of real numbers under each of the following functions?
   (a) f(x) = 4x + 2
   (b) f(x) = (x + 1)/2
   (c) f(x) = x² - (3/4) [difficult for all points; tell me about some points or ranges]
3. Compute the following complex numbers.
   (a) (3 + 2i) + (-7 + i)
   (b) (3 + 2i)(-7 + i)
   (c) (i/2)ⁿ for each positive integer n
   (d) (1 - i)ⁿ for each positive integer n
4. Compute f(f(1-i)) if f(z) = z² - 2.