should the domain be limited for squared fractions? I'm working through a practice exam and one question asks for the domain of
$\frac{1}{(x-4)^2}+3$
and says that the answer is $[1,4)$
I can see that there is an asymptote at $x=4$, and that the range is restricted because the fraction can only evaluate to positive numbers, but why isn't the domain just all real numbers excluding 4?
 A: One thing to note about functions is that they are only well-defined if it is clear what the domain is. This means that every time we define a new function $f$, we should really say something like

Consider the function $f:\mathbb{R}\mapsto\mathbb{R}$ defined by
$f(x)=x^2$ for all $x$.

This is too verbose for a working mathematician, and so this sentence often becomes

Consider the function $f$ defined by $f(x)=x^2$.

or simply

Consider the function $f(x)=x^2$.

While the latter two sentences are convenient, this convenience is at the cost of logical precision. The domain of a function is part of the definition of the function itself, and the function is $f$, not $f(x)$.
Because of the shorthand that mathematicians adopt, there is a strong convention that, unless otherwise specified, the domain of a function is the set of values for which a formula makes sense. Hence, people tend to say 'the domain of $x^2$ is $\mathbb{R}$'.
In your example, if no further information was given, then we would assume that the domain of the function $f$ given by
$$
f(x)=\frac{1}{(x-4)^2}+3
$$
is $x\neq4$. However, since the author has explicitly stated that the domain $[1,4)$, then the domain is quite clearly not $x\neq4$.
