To determine the correct statement about the value of [tex]\( m \)[/tex] such that the ranges of [tex]\( f(x) = \sqrt{m x} \)[/tex] and [tex]\( g(x) = m \sqrt{x} \)[/tex] are the same, let's analyze the given functions step-by-step:
1. Functions Analysis:
- [tex]\( f(x) = \sqrt{m x} \)[/tex]
- [tex]\( g(x) = m \sqrt{x} \)[/tex]
2. Determining the Range of [tex]\( f(x) \)[/tex]:
- For [tex]\( f(x) = \sqrt{m x} \)[/tex], we need [tex]\( m x \geq 0 \)[/tex].
- Since [tex]\( x \geq 0 \)[/tex] (as the domain of the square root function is non-negative numbers), [tex]\( m x \geq 0 \)[/tex] is satisfied if [tex]\( m \geq 0 \)[/tex].
- The range of [tex]\( f(x) \)[/tex] is when [tex]\( \sqrt{m x} \geq 0 \)[/tex], which is valid for all [tex]\( y \geq 0 \)[/tex].
3. Determining the Range of [tex]\( g(x) \)[/tex]:
- For [tex]\( g(x) = m \sqrt{x} \)[/tex], we again need [tex]\( x \geq 0 \)[/tex].
- The function [tex]\( m \sqrt{x} \)[/tex] will take values such that if [tex]\( m \geq 0 \)[/tex], [tex]\( g(x) \geq 0 \)[/tex] for [tex]\( x \geq 0 \)[/tex]. Conversely, if [tex]\( m < 0 \)[/tex], [tex]\( g(x) \leq 0 \)[/tex] for [tex]\( x \geq 0 \)[/tex].
4. Comparing the Ranges:
- If [tex]\( m \geq 0 \)[/tex], [tex]\( f(x) = \sqrt{m x} \)[/tex] and [tex]\( g(x) = m \sqrt{x} \)[/tex] should both be non-negative.
- If [tex]\( m < 0 \)[/tex], the range of [tex]\( g(x) = m \sqrt{x} \)[/tex] will be non-positive (all values less than or equal to zero), which cannot match the non-negative range of [tex]\( f(x) = \sqrt{m x} \)[/tex].
5. Finding Specific [tex]\( m \)[/tex]:
- For [tex]\( m = 1 \)[/tex]:
- [tex]\( f(x) = \sqrt{1 \cdot x} = \sqrt{x} \)[/tex]
- [tex]\( g(x) = 1 \cdot \sqrt{x} = \sqrt{x} \)[/tex]
- Clearly, both functions have the range [tex]\( y \geq 0 \)[/tex].
- For [tex]\( m \neq 1 \)[/tex]:
- Suppose [tex]\( m \neq 1 \)[/tex]. The function values will then depend on [tex]\( m \)[/tex] which will either stretch or compress the function differently, altering the range.
- To have the ranges exactly match, [tex]\( m \)[/tex] must be set such that [tex]\( f(x) = g(x) \)[/tex]. This is only satisfied if the coefficient in front of the square root in both functions is identical, that is when [tex]\( m = 1 \)[/tex].
Therefore, the value of [tex]\( m \)[/tex] that makes the range of [tex]\( f(x) \)[/tex] equal to the range of [tex]\( g(x) \)[/tex] can only be:
[tex]\[
m \text{ can only equal } 1.
\][/tex]
So, the correct statement is:
- [tex]\( m \)[/tex] can only equal 1.
