how can i define E, a set of linear function from $A\subseteq \mathbb{R}$ to $\mathbb{R}$ in a set notation form?

Question

My attempt is : $E = \{f\ |\ f(x) = ax+b, a,b\in \mathbb{R};\ x\in A\}$

Answer 1

It's definitely a good attempt, but here's a slightly more rigorous one: $$E = \lbrace f : A \to \mathbb{R} \mid \exists a, b \in \mathbb{R}, \forall x \in A, \; f(x) = ax + b \rbrace,$$ or indeed $$E = \lbrace f : A \to \mathbb{R} \mid \text{$f$ is a polynomial of degree at most $1$} \rbrace.$$ You are allowed to use words!