# Calculus

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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 ð while your friend Patrick’s is 3 and Karen’s is sin600 . 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.

Assume that the moving companies are described by the functions
f(x) = 5x and g(x) = 1/x

a. Find a general form for all the functions that you can get by taking
repeated compositions of the two functions.

b. What are all the possible addresses that you can move to? If the list is
infinite, list at least three possible addresses.

I'd be able to find b. if I could just understand a. I'm not quite sure what they're asking for when they want a "general form for all the functions" that I can get from repeated compositions of the two. Just one new function that serves for repeated compositions of both? Or something else entirely?

• Calculus -

Repeated compositions seems a bit vague. One way to interpret that is:

(fog)(x) = f(g(x)) = f(1/x) = 5/x

(fog)(5/x) = f(g(5/x)) = f(x/5) = x

So, the functions just swap back and forth between 5/x and x

That doesn't seem too useful.

Another way might mean

f(x) = 5x
f(g(x)) = f(1/x) = 5/x
f(g(f(x))) = f(g(5x)) = f(1/5x) = 1/x

So, now we see that fogof = g

I think you are right in considering that they want a single function, which should follow a pattern of repeated compositions, but I don't see anything useful.