How can we create a FunctionQ function?

Question

How can we check if an entry to our function is a Function or a Pure Function?

For example:

FunctionQ[target_]:=...

FunctionQ[Function[x, x^2]] == True
FunctionQ[#^2&] == True
FunctionQ[x^2] == False

It looks like we don't have a FunctionQ method available on Mathematica.

The motivation is that I'd like to combine this with the answer of this question to create only one function for the accepted answer of that link.

Edit:

Combining the provided answer, it looks like we can do the following:

FunctionQ[expression_Function?System`Private`ValidQ] := True;
FunctionQ[___] := False;
Test[expression_, variable_Symbol] :=
  Block[
    {final},
    If[
     FunctionQ[expression],
     final = expression,
     final = Function[variable, expression]
     ]
    ][4];

Test[Function[x, x^2], x] (* 1 *)

Test[#^2 &, x] (* 2 *)

Test[x^2, x] (* 3 *)

f1 = x^2; Test[f1, x] (* 4 *)

f2[x_] := x^2; Test[f2, x] (* 5 *)

Test[f2[x], x] (* 6 *)


16 (* 1 *)

16 (* 2 *)

16 (* 3 *)

16 (* 4 *)

b (* 5 *)

16 (* 6 *)

Answer 1

Observing the fact that valid Function objects seem to be marked as ValidQ we can use

FunctionQ[f_Function?System`Private`ValidQ] := True
FunctionQ[___] := False

Then use it via

In[100]:= FunctionQ[Function[{x, y}, x + 2]]

Out[100]= True

In[103]:= FunctionQ[# + 1 &]

Out[103]= True

In[105]:= FunctionQ[Log]

Out[105]= False

In[106]:= FunctionQ[Function[]]

During evaluation of In[106]:= Function::argb: Function called with 0 arguments; between 1 and 3 arguments are expected.

Out[106]= False