Testing properties of mapping given graph Suppose we are given a graph of a mapping (in $\Bbb R^2$ or $\Bbb R^3$) and we want to test graphically if it is a graph of a function, if it is one-to-one and if it is onto.


$\Bbb R^2$ : $$\begin{align} 
\text{Function} &amp; \iff \text{all lines parallel to the y-axis intersect the graph no more than once}.\\
\text{one-to-one}&amp; \iff \text{all lines parallel to the x-axis intersect the graph no more than once.}\\
\text{onto}&amp; \iff \text{all lines parallel to the x-axis intersect the graph at least once.}\\
\end{align}$$





$\Bbb R^3$ : $$\begin{align} 
\text{Function} &amp; \iff \text{all lines parallel to the z-axis intersect the graph no more than once}.\\
\text{one-to-one}&amp; \iff \text{all planes parallel to the xy-plane intersect the graph no more than once.}\\
\text{onto}&amp; \iff \text{all planes parallel to the xy-plane intersect the graph at least once.}\\
\end{align}$$


Are these methods correct?
In the case of Function, change "no more than once" to "exactly once" (assuming the function is defined on all of $\mathbb R$ or $\mathbb R^2$ respectively.
In the case of Function, change "no more than once" to "exactly once" (assuming the function is defined on all of $\mathbb R$ or $\mathbb R^2$ respectively.