Does $f\colon x\mapsto 2x+3$ mean the same thing as $f(x)=2x+3$? In my textbook there is a question like below:
If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$
As a multiple choice question, it allows for the answers:  
A. $11$
B. $5$
C. $\frac{1}{11}$
D. $9$
If what I think is correct and I read the equation as:
$$f(x)=2x-3$$
then,
$$y=2x-3$$
$$x=2y-3$$
$$x+3=2y$$
$$\frac {x+3} {2} = y$$
therefore:
$$f^{-1}(7)=\frac {7+3}{2}$$
$$=5$$

Answering your (main) question: Yes your interpretation of $f:x\rightarrow 2x-3$ being the same as $f(x)=2x-3$ is correct.
For your calculations: $\frac{7+3}{2}\neq10$.
