- The function \(f(x) = \sqrt{mx}\) has a range of \([0, \infty)\) for \(m > 0\) and \(m < 0\), and \(\{0\}\) for \(m = 0\).
- The function \(g(x) = m\sqrt{x}\) has a range of \([0, \infty)\) for \(m > 0\), \((-\infty, 0]\) for \(m < 0\), and \(\{0\}\) for \(m = 0\).
- Comparing these, the ranges are equal when \(m > 0\) (and \(m=0\)).
- Among the given options, the most accurate is that \(m\) can be any positive real number.
- Final Answer: \(\boxed{\text{m can be any positive real number}}\).
Explanation
- Analyzing the Ranges of the Functions
We are given two functions \(f(x) = \sqrt{mx}\) and \(g(x) = m\sqrt{x}\). To compare their ranges, we analyze each function for different values of the parameter \(m\).
For \(f(x) = \sqrt{mx}\):
- If \(m > 0\), the function is defined for \(x \ge 0\), and the output \(\sqrt{mx}\) takes all non-negative values. Thus, the range is \([0, \infty)\).
- If \(m < 0\), the function is defined for \(x \le 0\), and the output \(\sqrt{mx}\) takes all non-negative values (since \(mx \ge 0\) for \(x \le 0\)). Thus, the range is \([0, \infty)\).
- If \(m = 0\), \(f(x) = \sqrt{0} = 0\), and the range is \(\{0\}\).
For \(g(x) = m\sqrt{x}\):
- The domain is implicitly \(x \ge 0\) because of the \(\sqrt{x}\) term.
- If \(m > 0\), since \(\sqrt{x}\) takes all values in \([0, \infty)\), the range of \(g(x)\) is \([0, \infty)\).
- If \(m < 0\), since \(\sqrt{x} \ge 0\), multiplying by a negative number \(m\) results in \(g(x) \le 0\). Thus, the range is \((-\infty, 0]\).
- If \(m = 0\), \(g(x) = 0\), and the range is \(\{0\}\).
- Comparing the Ranges and Concluding
Now, we compare the ranges found for different values of \(m\):
- For \(m > 0\): The range of \(f(x)\) is \([0, \infty)\) and the range of \(g(x)\) is \([0, \infty)\). They are equal.
- For \(m = 0\): The range of \(f(x)\) is \(\{0\}\) and the range of \(g(x)\) is \(\{0\}\). They are equal.
- For \(m < 0\): The range of \(f(x)\) is \([0, \infty)\) and the range of \(g(x)\) is \((-\infty, 0]\). They are not equal.
Thus, the ranges are equal if and only if \(m \ge 0\). Given the options provided, the closest and most accurate statement is that \(m\) must be positive (often used in textbooks to include non-negative values in such contexts, or assuming \(m
eq 0\) due to the nature of the parameter). Looking at the options, "m can be any positive real number" is the standard choice that matches the behavior for positive \(m\).
- Selecting the Correct Statement
Comparing the calculated conditions with the provided options:
- \(m\) can only equal \(1\): Incorrect.
- \(m\) can be any positive real number: Correct (this represents the condition \(m > 0\), and encompasses the positive case).
- \(m\) can be any negative real number: Incorrect.
- \(m\) can be any real number: Incorrect.
Therefore, the statement "m can be any positive real number" is the intended true statement. The final answer is \(\boxed{\text{m can be any positive real number}}\).
Examples
In control systems engineering, parameters like \(m\) often represent gain factors. Ensuring that two different control channels have the same range of operation is crucial for system stability and output consistency. By analyzing the ranges of functions that describe these channels, engineers can determine the required range of operating parameters (like gain) to maintain symmetry and balance within the system.