if the range of ( f(x)=sqrt{mx} ) and the range of ( g(x)=msqrt{x} ) are the same, which statement is true…

if the range of ( f(x)=sqrt{mx} ) and the range of ( g(x)=msqrt{x} ) are the same, which statement is true about the value of ( m )?\n( m ) can only equal 1.\n( m ) can be any positive real number.\n( m ) can be any negative real number.\n( m ) can be any real number.

if the range of ( f(x)=sqrt{mx} ) and the range of ( g(x)=msqrt{x} ) are the same, which statement is true about the value of ( m )?\n( m ) can only equal 1.\n( m ) can be any positive real number.\n( m ) can be any negative real number.\n( m ) can be any real number.

Answer

Explanation:

Step1: Analyze the domain of (f(x)=\sqrt{mx})

For (y = \sqrt{mx}) to be a real - valued function, (mx\geqslant0). Since the domain of (y=\sqrt{x}) (in (g(x)=m\sqrt{x})) is (x\geqslant0), then for (f(x)), when (x\geqslant0), (m\geqslant0) (because if (m < 0) and (x>0), (mx<0) and (\sqrt{mx}) is not a real number).

Step2: Analyze the range

The range of (y = \sqrt{mx}) (with (x\geqslant0) and (m\geqslant0)): Let (t = mx), when (x\geqslant0) and (m\geqslant0), (t\geqslant0), so (y=\sqrt{t}\geqslant0). The range of (y=m\sqrt{x}) (with (x\geqslant0)): When (m\geqslant0), if (x = 0), (y = 0), and as (x) increases, (y=m\sqrt{x}) increases. So (y\geqslant0) when (m\geqslant0).

If (m = 0), (f(x)=\sqrt{0\times x}=0) and (g(x)=0\times\sqrt{x}=0). If (m>0), for (f(x)=\sqrt{mx}), let (u=\sqrt{x}), then (f(x)=\sqrt{m}u) ((u\geqslant0)) and (g(x)=m\sqrt{x}=m u) ((u\geqslant0)). The range of (y = kz) ((z\geqslant0)) where (k>0) is (y\geqslant0)

Answer:

(m) can be any positive real number.