Why is the correct value set of functions so arbitrary? When my professor describes a function, for example $f(x) = \sin(x)$, he is allowed to describe it as a function $f: A \rightarrow B$ with $A = B = \mathbb{R}$, even tough sin(x) only has values in $[-1, 1]$
However, when we are getting asked which is the value set of $f(x) = e^x$, the correct answer is not $\mathbb{R}$, but $\mathbb{R}_+$, because only positive values occur.
What do I miss here? I don't understand why sometimes you can be unprecise with your sets, and sometimes not.
 A: It is NOT wrong to say  $\sin(x): \mathbb{R} \rightarrow \mathbb{R}$ since [-1,1] is in $\mathbb{R}$ but perhaps the you could say "it is not honest" after we prove that $\sin(x)$ only takes on values in $[-1,1]$
Math generally only cares about correctness, and once you're correct you generally look for secondary goals of elegance and convenience so hence your professor doesn't bother restricting that set every time he/she/ze writes it. (Another example being $y = x^2: \mathbb{R} \rightarrow \mathbb{R}^+$ but usually they drop the $+$.)
Now while thats frustrating we could see it as motivation to create some new math. Given a TRUE mathematical sentence $S(s_1 ... s_n)$ which accepts sets $s_1 ... s_n$ as an input: we can say it is $\text{Minimal}$ if there doesn't exist an indexed set $s_i$ and another  $u$ such that $u \subset s_i$ but $S(s_1 ... u ... s_n)$ is also true.
To make this concrete, we let $s_1 = \mathbb{R}, s_2 = \mathbb{R}$. Then $S(s_1, s_2)$ is the statement $\sin(x): s_1 \rightarrow s_2$
The triple $\left( s_1 = \mathbb{R}, s_2 = \mathbb{R}, S(s_1, s_2) \right) $ is NOT minimal since as you noted $[-1,1] \subset \mathbb{R}$ that still lets $S$ be true so we can reduce the triple to:  $\left( s_1 = \mathbb{R}, s_2 = [-1,1] , S(s_1, s_2) \right) $.
Now this is minimal.
The general problem then of "minimizing" sentences can probably lead to some interesting in complex math. For example, what would it take to build an automated minimization program? Even for elementary sets this starts to be an interesting exercise in theorem proving and a hard software engineering challenge.
A: There is a difference between the range, or the image, of a function, and the 'codomain'.
When we write $f: A \rightarrow B$, this means that for every $a \in A$ there is a unique value $b \in B$ with $f(a) = b$.  However not every $b \in B$ needs to have an $a \in A$ with $f(a) = b$.  If the latter property is satisfied, $f$ is called surjective, or onto.  $B$ is called the codomain of $f$.  Sometimes the definition function includes a codomain, so that the codomain must be specified.  But mathematicians tend to be loose with this, so that we could write $\sin : \mathbb{R} \rightarrow \mathbb{R}$ just as well as $\sin : \mathbb{R} \rightarrow \mathbb{C}$.
On the other hand, the image, or the range, of $f$ is the set $\{b \in B: \exists a \in A \, f(a) = b\}$ is the set of all elements of $B$ that are actually mapped onto by $f$.  So the image of $\sin$ is $[-1,1]$, regardless of whether the codomain is considered to be $\mathbb{R}$ or $[-1,1]$.
I'm not sure about the definition of value set, so I can't say which definition of meant, but judging from the answer I assume your professor is asking about the range.
