Answer
$\mathrm{Remember:}\ $ We can consider vertical line test to see if the graph is a function of $\ x\ $ or not. If a vertical line, let's say $\ \ x=a\ \ $ intersects the graph at any two points, the graph would not be a function of $\ x.\ $
In both these cases, there exist a real number $\ a\ $ for which the vertical line $\ x=a\ $ intersects the graphs of $\ |y|=x\ $ and $\ |y|=|x|\ $ in two points. So, they both are not functions of $\ x.$
$\mathrm{See\:the\:graphs\:below.}$
Work Step by Step
$a).$ Graph of $\ \ |y|=x\ \ $ is the same as the graph of $\ \ y=|x|\ \ $ but with the switched $\ x\ $ and $\ y\mathrm{-axis}.$
$b).$ Graph of $\ \ y^2=x^2\ \ $is the same as the graph of $\ \ |y|=|x|.\ \ $ You can also graph functions $\ y=|x|\ $ and $\ y=-|x|\ $ combined to get the same graph.
