definition of a function: NOTATION CHECK Let $X,Y$ $\subset$ $\mathbb{R}$ where $x \in X$ and $f: X \to Y$. Would it be correct to say: 
"We may define what the map $f$ does by describing how all the domain elements are mapped to their images" (e.g. $f(x)=x^2$).
I just want to make sure this is the correct way to word what a function does. 
 A: I think that your definition is fine. Although, I will add some more detail to the definition of a function.
A function is a mapping from the set $X$ (the domain) into a set $Y$ (the range).
It maps every element $x\in X$ to an element $y \in Y$. Often the mapping is given through a formula:
$$f:X\to Y, ~f(x)=x^2$$
What is written above is the official way to specify a function: the domain, range, and mapping formula are explicitly specified. As suggested through a comment, you could also write $x\mapsto x^2$ instead of $f(x)=x^2$. In some textbooks and course notes, it is often assumed that the domain and range are implicitly understood $(X,Y\subset\mathbb R)$ and we write $f(x)=x^2$. The domain is everything where the formula makes sense and the range is where the formula takes the domain to.
Observe that we don't need to define a function through a formula such as $f(x)=x^2$ or $f(x)=2x+3$. We could also define a function $f:X\to Y$ by
$$f(2) = 7,~f(3) = 1,~f(5) = 11,~f(7) = 3,~f(11) = 9$$
or a function $g:[0,1] \to (\{0\},\{1\})$ by
$$g(x)=\begin{cases} 
      0 & \text{if x is rational} \\
      1 & \text{otherwise}
   \end{cases}
$$
It is important to understand that a function assigns to every element of the domain exactly one element of the range. This restriction isn't placed on the range. A function can map two different elements of the domain to the same element of the range.
