What are use-cases of non-surjective total functions? Total functions are so "interesting" in Mathematics that the term "total" is often implicit. Therefore, it is common to use "function" to refer to a "total function" and only mention "partial function" or "potentially partial function" if the special need arises.
My question is, why don't surjective functions get the same benefit? For every function $f: A \to B$ there exists a unique function $f_s: A \to Im_f(A)$ such that $\forall x \in A. f(x) = f_s(x)$. And that function is surjective. For all purposes that I can think of, $f_s$ can do the job just fine. Is there any great loss in expressivity for common use cases that I am overlooking which warrant talking about non-surjective functions by default? What would be those use-cases?
Edit: I am not trying to challenge the status quo. I am just trying to find examples which emphasize the interest in the non-surjectivity of a function.
 A: There are tons of situations where you care about functions that need not be surjective (but where you do care about their codomain).  Here are just a couple examples.
Given a set $X$, a metric on $X$ is a function $d:X\times X\to[0,\infty)$ which satisfies certain axioms that say that $d(x,y)$ behaves like the "distance" from $x$ to $y$.  A set $X$ together with a metric $d$ on $X$ is called a metric space.  Metric spaces play a huge role in modern mathematics and are ubiquitous in analysis and topology.  And there are plenty of natural and important examples of metrics which are not surjective.  For instance, if you take $X=[0,1]$ and $d(x,y)=|x-y|$, the image of $d$ is just $[0,1]$, rather than all of $[0,\infty)$.  If all functions were surjective, you would have to awkwardly define a metric to be a function from $X\times X$ to some subset of $[0,\infty)$, when that subset is usually of little interest.
Given a topological space $X$ and a point $x_0\in X$, the fundamental group of $X$ at $x_0$ is the set of continuous functions $f:[0,1]\to X$ such that $f(0)=f(1)=x_0$, modulo a certain equivalence relation (roughly, $f$ and $g$ are equivalent if you can "continuously deform" $f$ into $g$).  As the name suggests, this set has a natural group structure, and this group is hugely important in topology and geometry.  It is essential here that the codomain of $f$ is the space $X$ (the idea is that $f$ represents a "curve" drawn in the space $X$), but $f$ is not required to be surjective.  The definition of the equivalence relation also involves certain maps with codomain $X$ which are not required to be surjective.
The common theme in these examples is that we are not just talking about one particular function (for which you could always just change the codomain to make it surjective), but we are interested in studying an entire class of functions with some common properties.  One of the important common properties is that these functions take value in a certain set (such as $[0,\infty)$, or our chosen topological space $X$), but there is no reason for them to be surjective onto that set and they may have different images.  By allowing functions to not be surjective we can just specify their common codomain without caring about what their image is.
