Let $\text{Aff}(\mathbb{R})$ be a set of bijections from $\mathbb{R}$ to $\mathbb{R}$ of the form $$f_{a,b}(x) = ax + b$$ with $a \in \mathbb{R}^*$ and $b \in \mathbb{R}$.
Let now $\text{T} (\mathbb{R})$ be the set of functions of the form $f(x) = x+ b$. Show that $\text{Aff}(\mathbb{R})$ is a group and that $\text{T}(\mathbb{R})$ is a normal sub-group of $\text{Aff}(\mathbb{R})$.
I had no issues doing this. To show that $\text{Aff}(\mathbb{R})$ is a group, I simply showed that it is a sub-group of bijections from $\mathbb{R}$ to itself with the composition law. To show that $\text{T}(\mathbb{R})$ is normal didn't either give much difficulties.
Now, I am asked to do this:
Show that there exists and isomorphism between $\text{Aff}(\mathbb{R})/ \text{T}(\mathbb{R})$ and $\mathbb{R}^*$.
And here I struggle to give any answer. I am sure I have to construct the function. To do so, I would need to understand how do the elements of $\text{Aff}(\mathbb{R})/ \text{R}(\mathbb{R})$ look exactly. But I don't see what they look like. And then, I have no idea how to go on to construct an isomorphism!