Do you need to find the domain of a function to prove injectivity? I teach math and have had this debate with a fellow teacher. I believe the domain of a function is really not important to determine if a function is or is not injective if there is no "real life" context.
What do you think? 
would you find the domain of a function like
$f(x)=1-\ln(2-e^{2x})$ 
to show if the function is injective? If so, why?
 A: By definition functions have domains, proving injectivity without considering the domain is something devoided of sense, the domain of a function is part of its essence.
In practice there can be times in which whatever the domain is, the proofs will look the same (if the functions are in fact injective), but they are necessarily different because the 'first' step in proving injectivy of a function is taking 'two' arbitrary elements on its domain. You can't do this if you don't focus on the domain.
This doesn't mean that you have to 'find' the domain of a function to prove it is injective. Taking your example and letting $D_f$ denote the domain of $f$, the aforementioned first step would just be: let $x,y\in D_f\,\ldots$
A: A function $f:\ A\to B$ is tantamount to a subset $G_f\subset A\times B$ having certain properties. In most cases $A$ and $B$ are clearly specified in advance, and one can then start right away to investigate whether $f$ is injective or not.
Very often $A$ and/or $B$ have to be surmised from the context. While the exact envisaged range $B$ (e.g., ${\mathbb R}$ or ${\mathbb C}$) is not relevant for injectivity, the exact domain $A$ certainly is.
In cases where the function $f$ is given as an expression, e.g., $$f(x):=\qquad
{x\over 1+x^2},\quad \cos x, \quad 1-\log\bigl(2-e^{2x}\bigr),\quad {\rm etc.},$$
the tacit understanding is the following: Consider as domain the set of all $x$ envisaged in the context (integer, real, complex, or otherwise) for which the expression can be evaluated without asking questions. In this sense $0$ would not belong to the domain of $x\to{\sin x\over x}$. If one wants to restrict  $f$ to a smaller domain $A$, say in order to make it injective, one has to specify this domain explicitly.
A: I agree with Christians and Gits point, that the domain is an essential part of a function. But the problem here is also one of terminology. Especially in education before university, no clear distinction is made between the term, which is used to describe the combination of more elementary function to a new one, and the function itself.
The term $x⋅x$ (or $x^2$ by convention) can describe a lot of functions depending on the domain one chooses for the free variable $x$. It is valid for any set $M$ endowed with an operation $M×M →M$.
In school math the default domain is usually taken to be $ℝ$ or its biggest subset on which the function is well-defined. This particular choice of domain does however not make the domain irrelevant!
Like MPW mentioned in his comments, the exponential function is only injective on $ℝ$, but not on the bigger domain $ℂ$ (or even on an arbitrary manifold with affine connection, in general), while $x^2$, although not injective on $ℝ$, is injective when restricted to a suitable subset.
So it is clear, that the domain is vital information to determine the injectivity of a function. The only way in which it is not important is the following:

If a function with domain $M$ (e.g. $ℝ$) is injective, it is also injective after being restricted to any subset of $M$.

