Q:

Definitions of a function, a one-to-one function and an onto function

These are the definitions as I understand them...  are they correct?
Function:  for every element $x$ in the domain, there is one and only one corresponding element $f(x)$ in the codomain  $\longrightarrow$ $f(x)$ is not unique
One-to-One Function: for every element $x$ in the domain, there is one and only one corresponding $f(x)$ in the codomain $\longrightarrow$ $f(x)$ is unique
Onto Function:  for every element $f(x)$ in the codomain, there is one and only one corresponding element $x$ in the domain $\longrightarrow$ $x$ is unique

A:

For the definition of a function, you need $f(x)$ to be unique. Otherwise, it might be the case that $x$ is sent to more than one value and then what is $f(x)$?
For the definition of one-to-one, you need that for each $y$ in the codomain there is only one $x$ such that $y = f(x)$. For example, $f(x) = x^2$ is not one-to-one because $y = 1 = f(-1)=f(1)$.
For the definition of onto, we don't require uniqueness. It is permissible to have more than one value in the domain mapping to the same value in the codomain. However, be careful and pay attention to the existence part of this definition. For example, $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^2$ is not onto because no value in the domain maps to negative numbers which are in the codomain. 
Hope this helps!

