Answer
Please see the graph.
Work Step by Step
Red line: $y\le x^{2}+1$
Blue line: $y\ge 4-x$
Green line: $x\ge0$
Purple line: $y\ge0$
To determine what side of the lines to shade, we pick the point $(2,2)$.
$y\le x^{2}+1$
$2\le 2^{2}+1$
$2 \le 4+1$
$ 2 \le 5 $ (true, so we shade the side of graph with the point)
$y\ge 4-x$
$2\ge 4-2$
$2 \ge 2$ (true, so we shade the side of the graph with the point)
$x\ge0$
$2\ge0$ (true, so we shade the side of graph with the point)
$y\ge0$
$2\ge0$ (true, so we shade the side of graph with the point)
The solution set to the system of inequalities is the overlap of the four graphs.
