How do I prove that this specific claim is onto?

Question

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function and define a new function $h: \mathbb{R} \rightarrow \mathbb{R}$ via $h(x) = 2f(x) + 3$. If $f$ is onto, the $h$ is also onto.

Answer 1

Let $x \in \mathbb{R}$ than we want to find $y \in \mathbb{R}$ such that $2f(y)+3=x$. We know that there exists $z \in \mathbb{R}$ such that $f(z) =\frac{x-3}{2}$. Define $y=z$ and its easy to see that the indeed $2f(z)+3=x$.