If the range of f\left(x\right)=\sqrt{mx} and the range of g\left(x\right)=m\sqrt{x} are the same, which statement is true about the value of m?

So we are given two functions:
[tex]f(x)=\sqrt{mx}\\g(x)=m\sqrt x[/tex]
The range of the function g is the following
 [tex]x\geq 0 [/tex].
The range of the function f is:
[tex]m\times x\geq 0 [/tex]
Since the two ranges are equation, we deduce that the value of m is positive. 
Otherwise the solution of the above inequality would be 
[tex]x\leq 0 [/tex]
which is not the same as the first inequality.