Answer to the Discussion Question

Oct 13, 2025 by windailys.com 52 views

In mathematics, functions are used to describe the relationship between variables. When we have two functions, understanding their relationship can provide valuable insights into the behavior of the variables involved. In this article, we will explore the relationship between two functions, $$f(x)=\sqrt{m x}$$ and $$g(x)=m \sqrt{x}$$, and determine the conditions under which their ranges are the same.

The Range of a Function

The range of a function is the set of all possible output values it can produce for the given input values. In other words, it is the set of all possible y-values that the function can produce for the given x-values. To determine the range of a function, we need to consider the possible values of the input variable and the corresponding output values.

The Range of $$f(x)=\sqrt{m x}$$

The function $$f(x)=\sqrt{m x}$$ involves the square root of the product of $$m$$ and $$x$$. Since the square root function is only defined for non-negative values, the product $$m x$$ must be non-negative. This means that either $$m$$ is positive and $$x$$ is non-negative, or $$m$$ is non-positive and $$x$$ is negative.

If $$m$$ is positive, then the range of $$f(x)$$ is the set of all non-negative real numbers, since the square root of a positive number is always non-negative. If $$m$$ is non-positive, then the range of $$f(x)$$ is the set of all non-positive real numbers, since the square root of a negative number is always non-positive.

The Range of $$g(x)=m \sqrt{x}$$

The function $$g(x)=m \sqrt{x}$$ involves the product of $$m$$ and the square root of $$x$$. Since the square root function is only defined for non-negative values, $$x$$ must be non-negative. The product $$m \sqrt{x}$$ will be non-negative if $$m$$ is positive, and non-positive if $$m$$ is non-positive.

If $$m$$ is positive, then the range of $$g(x)$$ is the set of all non-negative real numbers, since the product of a positive number and a non-negative number is always non-negative. If $$m$$ is non-positive, then the range of $$g(x)$$ is the set of all non-positive real numbers, since the product of a non-positive number and a non-negative number is always non-positive.

Comparing the Ranges of $$f(x)$$ and $$g(x)$$

To determine the conditions under which the ranges of $$f(x)$$ and $$g(x)$$ are the same, we need to consider the possible values of $$m$$ and the corresponding ranges of the two functions.

If $$m$$ is positive, then the range of $$f(x)$$ is the set of all non-negative real numbers, and the range of $$g(x)$$ is also the set of all non-negative real numbers. In this case, the ranges of the two functions are the same.

If $$m$$ is non-positive, then the range of $$f(x)$$ is the set of all non-positive real numbers, and the range of $$g(x)$$ is also the set of all non-positive real numbers. In this case, the ranges of the two functions are the same.

Conclusion

In conclusion, the ranges of $$f(x)=\sqrt{m x}$$ and $$g(x)=m \sqrt{x}$$ are the same if and only if $$m$$ is positive or non-positive. This means that $$m$$ can be any positive or non-positive real number, and the ranges of the two functions will be the same.

Answer to the Discussion Question

Based on the analysis above, the correct answer to the discussion question is:

B. $$m$$ can be any positive real number.

In our previous article, we explored the relationship between two functions, $$f(x)=\sqrt{m x}$$ and $$g(x)=m \sqrt{x}$$, and determined the conditions under which their ranges are the same. In this article, we will answer some frequently asked questions about the relationship between these two functions.

Q: What is the range of $$f(x)=\sqrt{m x}$$?

A: The range of $$f(x)=\sqrt{m x}$$ is the set of all non-negative real numbers if $$m$$ is positive, and the set of all non-positive real numbers if $$m$$ is non-positive.

Q: What is the range of $$g(x)=m \sqrt{x}$$?

A: The range of $$g(x)=m \sqrt{x}$$ is the set of all non-negative real numbers if $$m$$ is positive, and the set of all non-positive real numbers if $$m$$ is non-positive.

Q: When are the ranges of $$f(x)$$ and $$g(x)$$ the same?

A: The ranges of $$f(x)$$ and $$g(x)$$ are the same if and only if $$m$$ is positive or non-positive.

Q: Can $$m$$ be any real number?

A: No, $$m$$ cannot be any real number. If $$m$$ is negative, then the range of $$f(x)$$ is the set of all non-positive real numbers, and the range of $$g(x)$$ is the set of all non-negative real numbers. In this case, the ranges of the two functions are not the same.

Q: What is the value of $$m$$ if the ranges of $$f(x)$$ and $$g(x)$$ are the same?

A: The value of $$m$$ if the ranges of $$f(x)$$ and $$g(x)$$ are the same is any positive or non-positive real number.

Q: Can $$m$$ be 0?

A: No, $$m$$ cannot be 0. If $$m$$ is 0, then the function $$f(x)$$ is not defined, since the square root of 0 is not defined.

Q: What is the relationship between $$f(x)$$ and $$g(x)$$ if $$m$$ is positive?

A: If $$m$$ is positive, then the range of $$f(x)$$ is the set of all non-negative real numbers, and the range of $$g(x)$$ is also the set of all non-negative real numbers. In this case, the ranges of the two functions are the same.

Q: What is the relationship between $$f(x)$$ and $$g(x)$$ if $$m$$ is non-positive?

A: If $$m$$ is non-positive, then the range of $$f(x)$$ is the set of all non-positive real numbers, and the range of $$g(x)$$ is also the set of all non-positive real numbers. In this case, the ranges of the two functions are the same.

Conclusion

In conclusion, the relationship between $$f(x)=\sqrt{m x}$$ and $$g(x)=m \sqrt{x}$$ is complex, and depends on the value of $$m$$. If $$m$$ is positive or non-positive, then the ranges of the two functions are the same. However, if $$m$$ is negative, then the ranges of the two functions are not the same.