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# What causes a set of functions to form a group under composition?

In my textbook the following functions are given and are said to form a group under composition:

$f_1(x) = x$,

$f_2(x) = \frac{1}{x}$,

$f_3(x) = 1-x$,

$f_4(x) = \frac{1}{1-x}$,

$f_5(x) = \frac{x-1}{x}$,

$f_6(x) = \frac{x}{x-1}$

As I understand it, composition means simplifying all possibilities of $f_n(f_m(x))$, and then testing for the group checks.

I understand that they form a group due to the fact that they satisfy the four axioms (closure, associativity, identity, inverse), but wish to understand exactly what about the functions causes this.

The best way I can put this is, if you needed to create another set, where would you start? I assume $f_1(x)$ is the only possible Identity Element for composition of functions, is there something else about the given functions that cause them to form a group?

As of now, this is all I notice:

$f_1(x)^{-1} = f_2(x), f_3(x)^{-1}=f_4(x),f_5(x)^{-1}=f_6(x)$