Why have we made a function to be many to one and not one to many? We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even being many to one, but being explicitly one to one. Why have we defined functions to be like this?
 A: Suppose we had a "one to many" "function". Then that would have to imply that $f(x)$ could be anything from some (possibly infinite) set of possible values $\{ y_1, y_2, \cdots \}$. This is a problem, since we then have:
$$x = y ~ ~ \text{need not imply} ~ ~ f(x) = f(y)$$
And how would you even choose what $f(x)$ should be, based solely on $x$? There are two solutions:


*

*Add some more state $s$ to the "function" $f$ which determines what $x$ should be sent to, but then at this point you have a normal many to one (or one to one) function $f_s$ and are back to square one

*Accept that the application of $f$ is nondeterministic, at which point most of the properties of calculus and math in general that you take for granted simply fly out the window (want to solve an equation? tough luck, applying the same "function" to both sides need not preserve the equation!)
So that settles the problem of why we can't expect one to many "functions" to behave the same as what is called a function in mathematics (even in principle). This is not to say such objects are necessarily useless - perhaps they have been studied. But they are not called functions, because they behave very differently.

Now, it is possible that by "one to many" you mean "sends a value to a set of values" (as opposed to one of a set of values as previously - note the difference). This is perfectly valid, and that function is in fact not one to many, as it still sends each value to one particular set of values. Not convinced? How about the function $f$ which, given a positive integer $n$, returns the set of positive multiples of $n$? We have:
$$f(1) = \{ 1, 2, 3, \cdots \}$$
$$f(2) = \{ 2, 4, 6, \cdots \}$$
$$f(3) = \{ 3, 6, 9, \cdots \}$$
Notice that each of the sets on the right-hand-side can be thought of as a single object, and there is no problem determining which set $n$ should be sent to, for any integer $n \geq 1$. We get:
$$f(n) = \{ n, 2n, 3n, \cdots \}$$
And $f$ is, indeed, a function. It also happens to be one-to-one, as it never sends two integers to the same set, even though individual sets might share values, e.g. $f(1)$ contains $2$, but so does $f(2)$. Think about that. Functions don't need to send numbers to numbers, they can be made much more general.

Finally, $\sin$ is in fact a function. Who told you otherwise? It is clearly many to one as it is periodic but it is not "one to many" as $\sin$ always maps an input $x$ to a single value $\sin(x)$ (the sine of $x$).
And a function in the general sense need not be one-to-one, it can be many-to-one (as $\sin$ just above is). But depending on the context, people might need to work exclusively with one-to-one functions because of their properties, and so will tend shorten that to "function" for brevity (you will usually see something like "let $f$ be a one-to-one function... the function... this function..." or "we will refer to one-to-one functions as functions from here on").
I hope this helps.
