If you insist on domain symmetry, it should be part of the test:
With[{dn = -1 <= x <= 4},
ForAll[x, Equivalent[dn, dn /. x -> -x]] // Reduce]
Pitfall: written this way it gives True:
dn = -1 <= x <= 4;
ForAll[x, Equivalent[dn, dn /. x -> -x]]
I myself would investigate parity this way first:
f[x_] := 1/(a^x - 1) + 1/2;
Reduce[f[-x] == -f[x], x]
Reduce[f[-x] == f[x], x]
