- Is every bytestring a valid Ed25519 private key?
- How to retrieve a US DS-160 confirmation page
- Reentering USA from Mexico with Tourist Visa
- Is homeopathy verifiable?
- Is time more “real” than math and why?
- Is there a name for empanadas in japanese?
- “den Hals umdrehen”
- Why do we read vayechal twice on a fast?
- Shammai - receive everyone with a pleasant countenance
- Eat Meat prepared for shabos on the 9 days
- How to update the SharePoint list form access?
- How to reduce size of UsageAndHealth database?
- Reading Excel Data With Javascript
- How to deal with my 6 year old son is who “over attached” with a girl?
- Install 2 different brands of brake pads on front wheels
- 2005 2500hd 4X4 Chevrolet, after replacing the air bags
- 1997 Ford F150XL - Front lights work, reverse lights work…brake lights and turn signals don't work
- Problem translation of terms menu - Drupal 8.5.1 - Distribution VarBase
- How to add a pager at both the top and bottom of a View
- Toggle button status (enabled / disabled) using ajax drupal form

# 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:

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