Question

If the range of f (x) = StartRoot m x EndRoot and the range of g (x) = m StartRoot x EndRoot are the same, which statement is true about the value of m?
m can only equal 1.
m can be any positive real number.
m can be any negative real number.
m can be any real number.

Answer 1

m can be any positive real number.

Reason: If m>0, sqrt(mx) = sqrt(m)·sqrt(x) and both f and g take all nonnegative values (range [0, ∞)). If m<0, f(x) (defined for x≤0) is nonnegative while g(x)=m·sqrt(x) is nonpositive, so their ranges differ. (m = 0 gives the trivial range {0}, but that choice is not listed.)