Given the function $f:[0,1]→[0,1]$; $f(x)=x^2$, check which one(s) of the properties it has. This homework is past due, but I am still fiddling trying to figure this out. 

question: I do not understand what the heck the notation of $f:[0,1] \to [0,1]$; means. I thought I did, but my repetition states otherwise.

Here is the image of the problem for better understanding

Don't even give me the answer to the question itself. Just please explain what f:[0,1] -> [0,1]; means. I thought it was all points $0 \le x \le1$. If it is, I have no idea what the heck the answer is. 

Any feedback or direction is appreciated.
It means the domain and codomain of the function are the closed interval from $0$ to $1$. Given any $x$ such that $0 \le x \le 1$, you must have $0 \le f(x) \le 1$.  This is true for $f(x)=x^2$.  You don't have to have that every point in $[0,1]$ is the image of some point in the domain.  If that is true, the function is surjective.
It is injective and surjective - both increasing and strictly increasing.
Graph: