Answer :
To determine which statement is true about the value of [tex]\( m \)[/tex] such that the range of [tex]\( f(x) = \sqrt{mx} \)[/tex] is the same as the range of [tex]\( g(x) = m \sqrt{x} \)[/tex], let's analyze the properties and domains of these functions step by step.
### Step-by-Step Analysis:
1. Understand the Functions:
- [tex]\( f(x) = \sqrt{mx} \)[/tex]
- [tex]\( g(x) = m \sqrt{x} \)[/tex]
2. Domain of the Functions:
- For [tex]\( f(x) = \sqrt{mx} \)[/tex], [tex]\( m \cdot x \)[/tex] needs to be non-negative (because we cannot take the square root of a negative number in the real number system). Hence, [tex]\( mx \geq 0 \)[/tex].
- For [tex]\( g(x) = m \sqrt{x} \)[/tex], [tex]\( \sqrt{x} \)[/tex] requires [tex]\( x \geq 0 \)[/tex], and [tex]\( m \)[/tex] can be any real number, affecting the range based on its sign.
3. Range Considerations:
- For [tex]\( f(x) \)[/tex], [tex]\( \sqrt{mx} \)[/tex] implies [tex]\( f(x) \geq 0 \)[/tex] if [tex]\( m \geq 0 \)[/tex]. If [tex]\( m < 0 \)[/tex], [tex]\( mx \)[/tex] would need [tex]\( x \leq 0 \)[/tex], which cannot align with [tex]\( \sqrt{x} \geq 0 \)[/tex].
- For [tex]\( g(x) \)[/tex], [tex]\( m \sqrt{x} \)[/tex] results in a range depending on the sign of [tex]\( m \)[/tex]. If [tex]\( m > 0 \)[/tex], then the range is [tex]\( [0, \infty) \)[/tex]. If [tex]\( m < 0 \)[/tex], the function produces non-positive values and needs [tex]\( x \geq 0 \)[/tex].
4. Equating the Ranges:
- For the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to match:
- If [tex]\( m = 1 \)[/tex], then both functions are exactly the same [tex]\( f(x) = g(x) = \sqrt{x} \)[/tex], and the range will be [tex]\( [0, \infty) \)[/tex].
- If [tex]\( m > 0 \)[/tex] and [tex]\( m \neq 1 \)[/tex], the ranges are still [tex]\( [0, \infty) \)[/tex] because the positive constant [tex]\( m \)[/tex] scales the input but does not affect the non-negativity.
- If [tex]\( m < 0 \)[/tex], the functions can't match in range because [tex]\( \sqrt{} \)[/tex] operation within [tex]\( f(x) \)[/tex] requires [tex]\( m \)[/tex] and [tex]\( x \)[/tex] to simultaneously allow non-negative values.
5. Conclusion:
- The value [tex]\( m \)[/tex] must be such that [tex]\( m > 0 \)[/tex], ensuring that both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the range [tex]\( [0, \infty) \)[/tex].
Given these observations, the correct statement is:
[tex]\( m \)[/tex] can be any positive real number.
### Step-by-Step Analysis:
1. Understand the Functions:
- [tex]\( f(x) = \sqrt{mx} \)[/tex]
- [tex]\( g(x) = m \sqrt{x} \)[/tex]
2. Domain of the Functions:
- For [tex]\( f(x) = \sqrt{mx} \)[/tex], [tex]\( m \cdot x \)[/tex] needs to be non-negative (because we cannot take the square root of a negative number in the real number system). Hence, [tex]\( mx \geq 0 \)[/tex].
- For [tex]\( g(x) = m \sqrt{x} \)[/tex], [tex]\( \sqrt{x} \)[/tex] requires [tex]\( x \geq 0 \)[/tex], and [tex]\( m \)[/tex] can be any real number, affecting the range based on its sign.
3. Range Considerations:
- For [tex]\( f(x) \)[/tex], [tex]\( \sqrt{mx} \)[/tex] implies [tex]\( f(x) \geq 0 \)[/tex] if [tex]\( m \geq 0 \)[/tex]. If [tex]\( m < 0 \)[/tex], [tex]\( mx \)[/tex] would need [tex]\( x \leq 0 \)[/tex], which cannot align with [tex]\( \sqrt{x} \geq 0 \)[/tex].
- For [tex]\( g(x) \)[/tex], [tex]\( m \sqrt{x} \)[/tex] results in a range depending on the sign of [tex]\( m \)[/tex]. If [tex]\( m > 0 \)[/tex], then the range is [tex]\( [0, \infty) \)[/tex]. If [tex]\( m < 0 \)[/tex], the function produces non-positive values and needs [tex]\( x \geq 0 \)[/tex].
4. Equating the Ranges:
- For the ranges of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] to match:
- If [tex]\( m = 1 \)[/tex], then both functions are exactly the same [tex]\( f(x) = g(x) = \sqrt{x} \)[/tex], and the range will be [tex]\( [0, \infty) \)[/tex].
- If [tex]\( m > 0 \)[/tex] and [tex]\( m \neq 1 \)[/tex], the ranges are still [tex]\( [0, \infty) \)[/tex] because the positive constant [tex]\( m \)[/tex] scales the input but does not affect the non-negativity.
- If [tex]\( m < 0 \)[/tex], the functions can't match in range because [tex]\( \sqrt{} \)[/tex] operation within [tex]\( f(x) \)[/tex] requires [tex]\( m \)[/tex] and [tex]\( x \)[/tex] to simultaneously allow non-negative values.
5. Conclusion:
- The value [tex]\( m \)[/tex] must be such that [tex]\( m > 0 \)[/tex], ensuring that both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the range [tex]\( [0, \infty) \)[/tex].
Given these observations, the correct statement is:
[tex]\( m \)[/tex] can be any positive real number.
