Is this expression allowed in a strict sense?
Consider a function $f:[a,b] \rightarrow [f(a),f(b)]$ defined by $f(x) = x$.
What I mean by it is that
Let $g:[a,b]\rightarrow \mathbb{R}$ defined by $ g(x) = x $. Consider $f:[a,b]\rightarrow g([a,b])$ where $f(x) = g(x)$ for all $x \in [a,b]$.
In summary, when the values of a function are explicitly defined, can I express the range of the function using its value? In the example, I wanted to make $f$ to be surjective.
I have not seen that notation used before In practice. However, I have seen similar notation used in topology to denote a chart map: where $M$ is a manifold and $U \subseteq M$ then let $x: U \to x(U)$ be a chart map such that $x(U) \subseteq \mathbb{R}^d$. One thing to keep in mind is the relation between the two sets, I.e., the map $x : U \to x(U)$ is always given before how the map is defined e.g. $x(t) := t^{2}$ $\forall t \in U$. With that in mind, the order at which you have presented the information about the map $f$ is a little unconventional.