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:
$$f^{-1}(7)=\frac {7+3}{2}$$
$$=5$$

As a matter of not merely style but of writing in complete sentences you should write $x=f^{-1}(7)\iff f(x)=7\iff 2x-3=7\iff (2x-3)+3=7+3\iff 2x=10\iff x=5.$ This  reduces errors and shows the flow of the reasoning. In this case the sequence of "$\iff$" shows that the implications go in both directions. The importance of this is that we can conclude that $x=5\implies f(x)=7$ and that $x\ne 5\implies f(x)\ne 7.$ There are other equations that  imply $x=5$ but in fact have no solutions. For example $x=x+1\implies x=5,$ but $x=5$ does not imply $x=x+1.$ 
