$\mathrm{with}\left(\mathrm{plots}\right)\&colon;$$\mathrm{inequal}\left(\left\{x^2-y<0\&comma;x^2\&plus;y^2<9\&comma;0<3y-x-2\right\}\&comma;x\&equals;-2..2\&comma;y\&equals;0..3\&comma;\mathrm{color}\&equals;''Niagara\; 1''\right)$

$\mathrm{ineqs}:=\left[\left[\frac{4}{9}<y\&comma;y<1\&comma;-\sqrt{y}<x\&comma;x<3y-2\right]\&comma;\left[1<y\&comma;y<\frac{3}{5}\&plus;\frac{1\sqrt{86}}{10}\&comma;-\sqrt{y}<x\&comma;x<\sqrt{y}\right]\&comma;\left[\frac{3}{5}\&plus;\frac{1\sqrt{86}}{10}<y\&comma;y<-\frac{1}{2}\&plus;\frac{1\sqrt{37}}{2}\&comma;-\sqrt{y}<x\&comma;x<\sqrt{y}\right]\&comma;\left[-\frac{1}{2}\&plus;\frac{1\sqrt{37}}{2}<y\&comma;y<3\&comma;-\sqrt{-y^2\&plus;9}<x\&comma;x<\sqrt{-y^2\&plus;9}\right]\right]\&colon;$

$\mathrm{i1}:=\mathrm{inequal}\left({\mathrm{ineqs}}_1\&comma;x\&equals;-2..2\&comma;y\&equals;0..3\&comma;\mathrm{color}\&equals;''Niagara\; 1''\right)\&colon;$$\mathrm{i2}:=\mathrm{inequal}\left({\mathrm{ineqs}}_2\&comma;x\&equals;-2..2\&comma;y\&equals;0..3\&comma;\mathrm{color}\&equals;''Niagara\; 2''\right)\&colon;$$\mathrm{i3}:=\mathrm{inequal}\left({\mathrm{ineqs}}_3\&comma;x\&equals;-2..2\&comma;y\&equals;0..3\&comma;\mathrm{color}\&equals;''Niagara\; 3''\right)\&colon;$$\mathrm{i4}:=\mathrm{inequal}\left({\mathrm{ineqs}}_4\&comma;x\&equals;-2..2\&comma;y\&equals;0..3\&comma;\mathrm{color}\&equals;''Niagara\; 4''\right)\&colon;$$\mathrm{display}\left(\mathrm{i1}\&comma;\mathrm{i2}\&comma;\mathrm{i3}\&comma;\mathrm{i4}\right)$

$\mathrm{inequal}\left(\mathrm{map}\left(t\&rarr;\left[t_{\left[\right]}\&comma;0\le y-2x-1\right]\&comma;\mathrm{ineqs}\right)\&comma;x\&equals;-2..2\&comma;y\&equals;0..3\&comma;\mathrm{optionsfeasible}\&equals;\left[\left[\mathrm{color}\&equals;''Niagara\; 1''\right]\&comma;\left[\mathrm{color}\&equals;''Niagara\; 2''\right]\&comma;\left[\mathrm{color}\&equals;''Niagara\; 3''\right]\&comma;\left[\mathrm{color}\&equals;''Niagara\; 4''\right]\right]\&comma;\mathrm{optionsopen}\&equals;\left[\mathrm{color}\&equals;''Mono\; 2''\&comma;\mathrm{thickness}\&equals;1\right]\&comma;\mathrm{optionsclosed}\&equals;\left[\mathrm{color}\&equals;''Mono\; 1''\&comma;\mathrm{thickness}\&equals;3\right]\&comma;\mathrm{optionsexcluded}\&equals;\left[\mathrm{color}\&equals;''Mono\; 5''\right]\right)$

$$\begin{gathered} \mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon; \\ \mathrm{DrawCayleyTable}\left(\mathrm{AlternatingGroup}\left(4\right)\&comma; \mathrm{colors}\&equals;\mathrm{ColorTools}\left[\mathrm{GetPalette}\right]\left(''generic''\right)\&comma; \mathrm{labels}\&equals;\mathrm{letters}\right) \end{gathered}$$

$\mathrm{DrawCayleyTable}\left(\mathrm{AlternatingGroup}\left(3\right)\&comma; \mathrm{colors}\&equals;\mathrm{ColorTools}\left[\mathrm{GetPalette}\right]\left(''resene''\right)\&comma; \mathrm{labels}\&equals;\mathrm{numbers}\right)$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_cuts}\&comma;\arcsin \left(2z\sqrt{1-z^2}\right)\&comma; \mathrm{plot}\&equals;\mathrm{`3D`}\right)$

$\left[\arcsin \left(2z\sqrt{-z^2\&plus;1}\right)\&comma;\mathrm{And}\left(z\&equals;-\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}-\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(1\&comma;\mathrm{\&infin;}\right)\right)\&comma;\mathrm{And}\left(z\&equals;-\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}-\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(-\mathrm{\&infin;}\&comma;-1\right)\right)\&comma;\mathrm{And}\left(z\&equals;-\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}\&plus;\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(1\&comma;\mathrm{\&infin;}\right)\right)\&comma;\mathrm{And}\left(z\&equals;-\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}\&plus;\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(-\mathrm{\&infin;}\&comma;-1\right)\right)\&comma;\mathrm{And}\left(z\&equals;\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}-\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(1\&comma;\mathrm{\&infin;}\right)\right)\&comma;\mathrm{And}\left(z\&equals;\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}-\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(-\mathrm{\&infin;}\&comma;-1\right)\right)\&comma;\mathrm{And}\left(z\&equals;\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}\&plus;\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(1\&comma;\mathrm{\&infin;}\right)\right)\&comma;\mathrm{And}\left(z\&equals;\frac{1}{2}\sqrt{1\&plus;\mathrm{\&alpha;}}\&plus;\frac{1}{2}\sqrt{1-\mathrm{\&alpha;}}\&comma;\mathrm{\&alpha;}\&in;\mathrm{RealRange}\left(-\mathrm{\&infin;}\&comma;-1\right)\right)\&comma;\mathrm{And}\left(\mathrm{\&Im;}\left(z\right)\&equals;0\&comma;1<\mathrm{\&real;}\left(z\right)\right)\&comma;z<-1\right]$

$\mathrm{plot3d}\left(\sin \left(x\right)\&plus;\cos \left(y\right)\&comma;x\&equals;-5..5\&comma;y\&equals;-5..5\right)$