If the range of [tex]f(x)=\sqrt{m x}[/tex] and the range of [tex]g(x)=m \sqrt{x}[/tex] are the same, which statement is true about the value of [tex]m[/tex]?
A. [tex]m[/tex] can only equal 1.
B. [tex]m[/tex] can be any positive real number.
C. [tex]m[/tex] can be any negative real number.
D. [tex]m[/tex] can be any real number.
Answer (2)
Analyze the ranges of f ( x ) = m x and g ( x ) = m x for 0"> m > 0 , m < 0 , and m = 0 .
When 0"> m > 0 , both functions have a range of [ 0 , ∞ ) .
When m < 0 , the ranges are different: f ( x ) has a range of [ 0 , ∞ ) , while g ( x ) has a range of ( − ∞ , 0 ] .
When m = 0 , both functions have a range of 0 . Therefore, m can be any positive real number.
The final answer is: m can be any positive real number .
Explanation
Analyzing the Problem Let's analyze the given functions and their ranges to determine the possible values of m . We have two functions: f ( x ) = m x and g ( x ) = m x . We want to find the values of m for which the ranges of these two functions are the same.
Case 1: m > 0 Case 1: 0"> m > 0 If m is positive, then for f ( x ) = m x to be a real number, m x must be non-negative. Since 0"> m > 0 , this means x ≥ 0 . Thus, the domain of f ( x ) is [ 0 , ∞ ) . The range of f ( x ) is also [ 0 , ∞ ) because as x varies from 0 to infinity, m x also varies from 0 to infinity. For g ( x ) = m x , the domain is x ≥ 0 , and since 0"> m > 0 , the range is [ 0 , ∞ ) .
In this case, the ranges of f ( x ) and g ( x ) are the same, i.e., [ 0 , ∞ ) .
Case 2: m < 0 Case 2: m < 0 If m is negative, then for f ( x ) = m x to be a real number, m x must be non-negative. Since m < 0 , this means x ≤ 0 . Thus, the domain of f ( x ) is ( − ∞ , 0 ] . The range of f ( x ) is [ 0 , ∞ ) because as x varies from 0 to negative infinity, m x varies from 0 to infinity. For g ( x ) = m x , the domain is x ≥ 0 . Since m < 0 , the range of g ( x ) is ( − ∞ , 0 ] .
In this case, the ranges of f ( x ) and g ( x ) are not the same. The range of f ( x ) is [ 0 , ∞ ) , while the range of g ( x ) is ( − ∞ , 0 ] .
Case 3: m = 0 Case 3: m = 0 If m = 0 , then f ( x ) = 0 ⋅ x = 0 and g ( x ) = 0 ⋅ x = 0 . The range of both functions is 0 . In this case, the ranges are the same.
Conclusion From the above analysis, the ranges of f ( x ) and g ( x ) are the same when 0"> m > 0 and when m = 0 . Therefore, m can be any non-negative real number. However, the options provided do not include 'any non-negative real number'. The closest option is 'm can be any positive real number'.
Final Answer Therefore, the correct statement is that m can be any positive real number.
Examples
Understanding the range of functions is crucial in various fields. For instance, in physics, when modeling projectile motion, the range of a function describing the projectile's height can tell us the maximum height the projectile can reach. Similarly, in economics, the range of a cost function can help determine the possible cost outcomes for a business. In computer graphics, understanding the range of color values ensures that colors are displayed correctly on a screen. These examples highlight how analyzing function ranges helps in making informed decisions and predictions in real-world scenarios.
The ranges of the functions f ( x ) = m x and g ( x ) = m x are the same when 0"> m > 0 or m = 0 . Therefore, the correct answer is that m can be any positive real number. Thus, the correct option is B.
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