f[x_] := 2 (x + 1)/(x + 2);

Nest[f, 1, 5]

N[%]

?? Nest

Nest[f, 1, 10]

N[%]

NestList[f, 1, 10]

NestList[f, -1, 10]

NestList[f, -5/3, 10]

2/(2 - Sqrt[2])^2  // N

FixedPoint[f, -1]

FixedPoint[f, 1]

g[x_] := x/2 + 1/x

NestList[g, 1, 10]

N[%, 50]

N[Sqrt[2], 50]

?? FindRoot

gr = Plot[f[x], {x, -1, 3}]

gr

Show[gr]

gr  // InputForm

?? Circle

Circle[{0, 1}, 2]

Graphics[%]

Show[%]

Show[Graphics[Circle[{0, 1}, 2]], 
      AspectRatio -> Automatic]

Show[Graphics[Disk[{0, 1}, 2]], 
      AspectRatio -> Automatic]

Show[Graphics[Line[{{0, 0}, {2, 3}}]], Axes -> True]

Show[Graphics[Line[{{0, 0}, {2, 3}, {5, 1}, {2, -1}, {0, 0}}]], 
      Axes -> True]

Show[Graphics[Polygon[{{0, 0}, {2, 3}, {5, 1}, {2, -1}, {0, 0}}]], 
      Axes -> True]

f[x_] := Sin[x] + 2  x - y

f'[x]

e = 10^(-10);
h[y_] := 
      Module[{a, x}, 
        For[a = 2  y; i = 0, i < 15, i++, 
          x = a - (Sin[a] + 2  a - y)/(2 + Cos[a]); 
          If[Abs[x - a] < e/10, Return[x], a = x]]]

h[3. ]

Plot[Sin[h[y]] + 2  h[y] - y, {y, 0, 1}]

a