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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: $\frac{7+3}{2}\neq10$.
This is to just to elaborate on why someone would use the notation $$f: x \mapsto 2x-3$$ When we treat a function $f$ as an object, i.e. do more with it than just evaluate it at points, then we need to be able to distinguish between the object, the function $f$, and its rule of assignment determining its values at points $x$, i.e. after evaluating at points $x$.
Since $f(x)$ denotes both the function $f$, as an object in its own right, as well as the value of that function evaluated at a point $x$, it is too ambiguous for such purposes, because it does not allow us to distinguish between the function and its rule of assignment.
Often we can identify the function $f$, as an object in its own right, with its rule of assignment taking $x$ to $f(x)$, since we are in such instances only considering the latter, so no ambiguity arises.
However, the notation $$f:x\mapsto 2x-3$$ is nice because it both presents the function $f$ as a distinct object while also specifying its rule of assignment. The colon : is meant to be read "such that", which implies that the expression $f:x\mapsto 2x-3$ reads "the function $f$ such that $x$ is mapped to $2x-3$".
The benefit of this notation is that it allows us to distinguish between the function $f$ and its rule of assignment when we need to distinguish between the two, while taking exactly as long to write as $f(x)=2x-3$, when we don't need to distinguish between the function and its rule of assignment, and thus economy of notation becomes a priority.
Thus, the notation $f:x\mapsto2x-3$ is both exactly as efficient as the classical notation $f(x)=2x-3$ and less ambiguous.
(note: this answer was formulated in response to the original version of the question, which has since been edited by a third party)
You're parsing the expression wrong; it's not
$$ \color{red}{\mathbf{ f : x }} \mapsto 2x-3 $$
instead, it is
$$ f : \color{red}{\mathbf{ x \mapsto 2x-3 }}$$
That is, $x \mapsto 2x-3$ is one particular notation for "the function which, for all $x$, sends the value $x$ to the value $2x-3$". The colon notation is one particular way to attach a name to the mapping; to say "$f$ is the function which, for all $x$, ...". You might read it as "$f$ sends $x$ to $2x-3$".
The colon notation is more often used when you include a third term expressing the domain and codomain. e.g. if we are talking about a real-valued function on the reals, we would wrote
$$ f : \mathbb{R} \to \mathbb{R} : x \mapsto 2x-3 $$
Here:
Your mistake is here: $$ x-3=2y $$ It should be $$ x+3=2y $$
Interpret $$f:\quad x\mapsto y:=2x-3$$ as a flow diagram: The operation $f$ takes a variable value $x$ as input and produces a variable value $y$ as output, whereby the exact formula for computing $y$ from $x$ is given.
I'm writing here because in your argument you replace without hesitation $y=2x-3$ with $x=2y-3$. This seems to be a recommended procedure, but is a terrible mistake, because now both $x$ and $y$ have two different meanings in the same chain of reasoning. Keep the names of the variables as long as you are looking at $f$ and $f^{-1}$ simultaneously.
Argue as follows instead: The inverse function $f^{-1}$ should produce the required input value $x$ for a given output value $y$ of $f$. We therefore have to solve the equation $y=2x-3$ for $x$ "as a function of $y$". The result of course is $x={1\over2}(y+3)$, so that the flow diagram for $f^{-1}$ looks as follows: $$f^{-1}:\quad y\mapsto x:={1\over2}(y+3)\ .$$ In particular $f^{-1}(7)=5$.
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 $x=x+1.$
$\frac{1}{11}$. Also, welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? $\endgroup$ – 5xum Nov 8 '16 at 10:06