f[x_] := Tan[Sqrt[x]]/Sqrt[x]
f[0] = f[0.] = Limit[f[x], x -> 0]
(* 1 *)
FunctionDomain[f[x], x]
(* NotElement[1/2 + Sqrt[x]/π, Integers] && x > 0 *)
s = x /. Solve[1/2 + Sqrt[x]/π == 1, x][[1]]
(* π^2/4 *)
if[y_?NumericQ] := x /. NSolve[{f[x] == y, 0 <= x < 10}, x][[1]]
Plot[if[y], {y, -5, 5},
AxesLabel -> (Style[#, 14, Bold] & /@ {y, x}),
Epilog -> {Gray, Dashed, Line[{{-5, s}, {5, s}}]}]

EDIT:
$Version
(* "11.2.0 for Mac OS X x86 (64-bit) (September 11, 2017)" *)
if
(which is to say NSolve
) fails for some values
if[1.00001]
(* Part::partw: Part 1 of {} does not exist.
ReplaceAll::reps: {{}[[1]]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing.
x /. {}[[1]] *)
Redefine f
as
f2[x_] = Sinc[Sqrt[x]]/Cos[Sqrt[x]];
f
and f2
are equivalent except that f2[0]
is defined without adding a separate definition.
f[x] == f2[x] // FunctionExpand
(* True *)
f2[0]
(* 1 *)
if2[y_?NumericQ] :=
x /. NSolve[{f2[x] == y, 0 <= x < 10}, x][[1]]
Plot[if2[y], {y, -5, 5},
AxesLabel -> (Style[#, 14, Bold] & /@ {y, x}),
Epilog -> {Gray, Dashed, Line[{{-5, s}, {5, s}}]}]

if2[1.00001]
(* 0.0000299996 *)
f[x_]:=Tan[Sqrt[x]/Sqrt[x]
is not well defined, as it has a syntax error :) $\endgroup$