p1 = Plot[2/(1 + 20 x^2), {x, 0, 3}, AxesLabel -> {"x", "y"},
  LabelStyle -> (FontSize -> 16), GridLines -> Automatic,
  PlotRange -> {0, 3}]

p2 = Plot[1/(1 + 20 x^2)^(1/2), {x, 0, 3}, AxesLabel -> {"x", "y"},
  LabelStyle -> (FontSize -> 16), GridLines -> Automatic,
  PlotRange -> {0, 3}]

Show[p1, p2]

GraphIntersection[p1, p2]

f1 = 2/(1 + 20 x^2);
f2 = 1/(1 + 20 x^2)^(1/2);

xp = FindInstance[f1 == f2 && 0 < x < 3, x, Reals, 15] // Values // Flatten

yp = f1 /. x -> xp

Plot[{f1, f2}, {x, 0, 3},
 AxesLabel -> {"x", "y"},
 Epilog -> {Red, PointSize[0.02], Point[{First@xp, First@yp}]},
 LabelStyle -> (FontSize -> 16),
 GridLines -> Automatic,
 PlotRange -> {0, 3},
 PlotTheme -> "Detailed"]

ClearAll[f, g];
f = 2/(1 + 20 x^2);
g = 1/(1 + 20 x^2)^(1/2);
Plot[{f, g}, {x, 0, 3},
 MeshFunctions -> {(f - g) /. x -> # &}, Mesh -> {{0}},
 MeshStyle -> Directive[Red, PointSize[Large]],
 AxesLabel -> {"x", "y"}, LabelStyle -> (FontSize -> 16),
 GridLines -> Automatic, PlotRange -> {0, 3}]

ClearAll[f2, g2, x];
f2 = 3/(3 + 20 (Sin@x - 1/2)^2);
g2 = 1/(1 + 5 (Sin[x] - 3/10)^2)^(1/2);
Plot[{f2, g2}, {x, 0, 3}, MeshFunctions -> {(f2 - g2) /. x -> # &},
 Mesh -> {{0}}, MeshStyle -> Directive[Red, PointSize[Large]],
 AxesLabel -> {"x", "y"}, LabelStyle -> (FontSize -> 16),
 GridLines -> Automatic, PlotRange -> {0, 3/2}]

mesh = Last @@@ N[Solve[{f2 == g2, 0 <= x <= 3}, x, Reals] ]
(* {2.7032,0.438392,2.29184,0.849753} *)

Plot[{f2, g2}, {x, 0, 3}, Mesh -> {mesh},
 MeshStyle -> Directive[Red, PointSize[Large]],
 AxesLabel -> {"x", "y"}, LabelStyle -> (FontSize -> 16),
 GridLines -> Automatic, PlotRange -> {0, 3/2}]
(* same picture *)