rectplot[fff_,inputx_,inputy_, opts___] := ParametricPlot[Through[{Re,Im}[fff[x+I*y]]], {x,inputx[[1]], inputx[[2]]}, {y,inputy[[1]],inputy[[2]]}, opts, PlotStyle->None, Mesh->True];
discplot[fff_,inputx_,inputy_, opts___] :=ParametricPlot[Through[{Re,Im}[fff[r Exp[I t]]]],{r,inputx[[1]], inputx[[2]]},{t,inputy[[1]],inputy[[2]]},opts,PlotStyle->None, Mesh->True];

rectplot[Identity,{0,2},{0,2}]

f[z_] := (a z + b)/(c z + d)

a=2; b=3; c=-1; d=1; rectplot[f,{0,2},{0,2}]

t[z_] := ((z2-z3)/(z2-z1)) ((z-z1)/(z-z3))

z1=0; z2=2; z3=3; w1=1; w2=2; w3=3;

Clear[w1,w2,w3]

s[z_] := (((z-w1) (w2-w3))/((-w1+w2) (z-w3)))

s[z]

Solve[s[w]==z,w]

sinv[z_] := ((-w1 w2+w1 w3-w1 w3 z+w2 w3 z)/(-w2+w3-w1 z+w2 z))

t[z_] := z/3

sinv[t[z]]

Simplify[%]

fun[z_]:= 2((2-z)/(z-4))

fun[3]

t[z]

s[z]

sinv[t[z]]

FullSimplify[%]

s[w1]

sinv[1]

Simplify[%]

rectplot[t,{0,2},{0,2}]

discplot[Identity,{0,1},{0,Pi/2}]

discplot[f,{0,1},{0,Pi}]

discplot[s,{0,5},{0,2Pi}]