Problem understanding Group Theory question

[twotwo]

Hello all, my first post, hope to be a regular forum goer. Any help understanding this problem would be appreciated.

Homework Statement

"Consider the following functions: f(x) = 1/x ; g(x) = 1/(1-x) defined on the set R\{0,1} = (-∞,0) U (0,1) U (1,∞)

How many total functions can be generated by composing combinations of any number of these two functions?"

The Attempt at a Solution

What i am having trouble with is the word "combination". Does it mean any combination of adding, subtracting, multiplying and dividing? Or does it mean to take one function of another (as in, g(f(g(f(g(x)))))? I assume it means the latter, but that assumption comes merely from the limited number of functions.
Once again, any help would be immensely appreciated.

 

[Dick]

I think it means exactly what you think it means. It says "composing". I think the group operation is intended to be composition of functions.

 

[psholtz]

Yes, composing functions is what's intended..

For instance, if you take:

$$f(x) = \frac{1}{x}$$

$$f(f(x)) = x$$

$$f(f(f(x))) = \frac{1}{x}$$

So there are a total of 2 functions that can be created by composing f with itself (ad infinitum).

Continue mixing combinations of these two functions, and you'll get the total number of functions that can be created.