how do I prove all group properties?

Question

$G(x) = ax + b$ where $a$ is not equal to $0$ and $b$ are real numbers.

Prove $G$ is a group under function composition.

I understand that for every $a$ there is a corresponding $b$-value that does not repeat; however, I do not understand how I can prove the group properties.

Answer 1

First, $G(x) = ax +b$ is a function, not a set. Let $f_{a,b}$ be the function defined by $x \mapsto ax +b$. You want to show that

$$G = \{f_{a,b} \mid a,b \in \mathbb{R}, a \neq 0\}$$ is a group.

Normally, this would require you to check all group axioms, but because you know that the set of bijective functions (i.e. the functions that have an inverse), together with function composition, is a group, it suffices to show that $G$ is a subgroup of this group, for which you have to show $3$ things:

1) $G \neq \emptyset$ (hint: consider the map: $x \mapsto x)$

2) $f_{a,b}, f_{c,d} \in G \implies f_{a,b} \circ f_{c,d} \in G$ (hint: what function is $f_{a,b} \circ f_{c,d}$?)

3) $f_{a,b} \in G \implies {f_{a,b}}^{-1} \in G$ (hint: what function is ${f_{a,b}}^{-1}$?)

Answer 2

If you're more familiar with matrices, then the group is isomorphic to the set of $2\times 2$ matrices of the form $$ \pmatrix{ a & b \\ 0 & 1} $$ It is easy to see that function composition corresponds to matrix multiplication.