Determining the Value of m

Oct 13, 2025 by windailys.com 3 views

Understanding the Relationship Between Functions f(x) and g(x)

When dealing with functions, it's essential to understand the relationship between their domains and ranges. In this case, we're given two functions, f(x) and g(x), and we're asked to determine the value of m that makes their ranges the same.

The Function f(x)

The function f(x) is defined as f(x) = √(mx), where m is a positive real number. To find the range of this function, we need to consider the possible values of x that can be input into the function.

Domain of f(x): Since the square root of a negative number is undefined in the real number system, the domain of f(x) is restricted to non-negative values of x. Therefore, x ≥ 0.
Range of f(x): To find the range of f(x), we need to consider the possible values of √(mx). Since x ≥ 0, mx ≥ 0, and the square root of a non-negative number is always non-negative. Therefore, the range of f(x) is [0, ∞).

The Function g(x)

The function g(x) is defined as g(x) = m√x, where m is a positive real number. To find the range of this function, we need to consider the possible values of x that can be input into the function.

Domain of g(x): Since the square root of a negative number is undefined in the real number system, the domain of g(x) is restricted to non-negative values of x. Therefore, x ≥ 0.
Range of g(x): To find the range of g(x), we need to consider the possible values of m√x. Since x ≥ 0, m√x ≥ 0, and the range of g(x) is [0, ∞).

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

Since the range of f(x) is [0, ∞) and the range of g(x) is also [0, ∞), we can conclude that the ranges of f(x) and g(x) are the same.

Determining the Value of m

To determine the value of m that makes the ranges of f(x) and g(x) the same, we need to consider the following:

Case 1: m = 1: If m = 1, then f(x) = √x and g(x) = √x. In this case, the ranges of f(x) and g(x) are the same.
Case 2: m ≠ 1: If m ≠ 1, then f(x) = √(mx) and g(x) = m√x. In this case, the ranges of f(x) and g(x) are not the same.

Therefore, the only value of m that makes the ranges of f(x) and g(x) the same is m = 1.

Conclusion

In conclusion, the value of m that makes the ranges of f(x) and g(x) the same is m = 1. This is because when m = 1, the functions f(x) and g(x) are equivalent, and their ranges are the same.

Final Answer

The final answer is A. m can only equal 1.
Q&A: Understanding the Relationship Between Functions f(x) and g(x)

In our previous article, we explored the relationship between the functions f(x) = √(mx) and g(x) = m√x, and we determined that the value of m that makes their ranges the same is m = 1. In this article, we'll answer some frequently asked questions about this topic.

Q: What is the significance of the value m = 1?

A: The value m = 1 is significant because it makes the functions f(x) and g(x) equivalent. When m = 1, f(x) = √x and g(x) = √x, and their ranges are the same.

Q: Can m be any positive real number?

A: No, m cannot be any positive real number. The value of m must be 1 in order for the ranges of f(x) and g(x) to be the same.

Q: What happens if m ≠ 1?

A: If m ≠ 1, then the ranges of f(x) and g(x) are not the same. The function f(x) = √(mx) has a range of [0, ∞), while the function g(x) = m√x has a range of [0, ∞) if m > 1, and a range of [0, m) if 0 < m < 1.

Q: How do the domains of f(x) and g(x) affect their ranges?

A: The domains of f(x) and g(x) are restricted to non-negative values of x, which means that x ≥ 0. This restriction affects the ranges of the functions, as the square root of a negative number is undefined in the real number system.

Q: Can the ranges of f(x) and g(x) be the same if m ≠ 1?

A: No, the ranges of f(x) and g(x) cannot be the same if m ≠ 1. The value of m must be 1 in order for the ranges of f(x) and g(x) to be the same.

Q: What is the relationship between the functions f(x) and g(x) when m = 1?

A: When m = 1, the functions f(x) and g(x) are equivalent, and their ranges are the same. In this case, f(x) = √x and g(x) = √x.

Q: Can the functions f(x) and g(x) be equivalent if m ≠ 1?

A: No, the functions f(x) and g(x) cannot be equivalent if m ≠ 1. The value of m must be 1 in order for the functions to be equivalent.

Conclusion

In conclusion, the value of m that makes the ranges of f(x) and g(x) the same is m = 1. This is because when m = 1, the functions f(x) and g(x) are equivalent, and their ranges are the same. We hope this Q&A article has helped to clarify any questions you may have had about this topic.

Final Answer

The final answer is A. m can only equal 1.