Imagine that you live on an infinitely long and straight street called Infinite Drive. The addresses on Infinite Drive are given by real numbers. Your address on the street is π while your friend Patrick’s is √3 and Karen’s is sin60°. There are two moving companies, f and g. Those companies move people along Infinite Drive from address x to new addresses f(x) or g(x), depending on which company people choose.

Question

1.Assume that the functions f(x) = 1-x and g(x) = 1/x. Describe how the moving companies move people from one address to another.

a.What are all possible addresses that you can move to when taking repeated compositions of the functions f and g?

b.How many possible addresses can you relocate to when using those moving companies? Are they finite? If so, make a list of all the possible new addresses, if your original address is π.
c.If your parents live on address 0, what are all the possible addresses that they can move to by repeatedly composing the two functions?

Answer 1

f(x) = 1-x
f(f(x)) = 1 - (1-x) = x

g(x) = 1/x
g(g(x)) = 1/(1/x) = x

So, in both cases, there are only two possible addresses available for each real number. Alas, g(0) is undefined, so it looks like the mover gets lost.