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 $\ |x|+|y|=1\ $ and $\ |x+y|=1\ $ in two points. So, they both are not functions of $\ x.$
$\mathrm{See\:the\:graphs\:below.}$
Work Step by Step
$a).$ When $\ x=0,\ |y|=1\ \rightarrow\ \ y=\pm1.\ $ When $\ y=0,\ |x|=1\ \rightarrow\ \ x=\pm1.\ $
So, we will have 4 different points for the given function. As we can see the given function is linear, just connect these 4 points with line segments.
$b).$ For the absolute value equation, we always get two solutions. These are $\ \ x+y=1\ \ \mathrm{and}\ \ -x-y=1.$ Just graph these lines.
