The average salary, in thousands of dollars, for science and engineering faculty is modeled by function g, where $x$ is the number of years since the manager began recording the data.
[tex]g(x)=15 \sqrt[3]{x-4}+40[/tex]
Which statement is true about the salaries of faculty members as $x$ increases?
A. As $x$ approaches positive infinity, the average salary for science and engineering faculty will always be greater than the average salary for arts and humanities faculty.
B. As $x$ approaches positive infinity, the average salary for science and engineering faculty will eventually be greater than the average salary for arts and humanities faculty.
C. As $x$ approaches positive infinity, the average salary for arts and humanities faculty will always be greater than the average salary for science and engineering faculty.
D. As $x$ approaches positive infinity, the average salary for arts and humanities faculty will eventually be greater than the average salary for science and engineering faculty.
Answer (2)
The function g ( x ) = 15 3 x − 4 + 40 models the average salary for science and engineering faculty.
As x increases, g ( x ) also increases without bound.
Regardless of how the average salary for arts and humanities faculty behaves, g ( x ) will eventually be greater.
Therefore, as x approaches positive infinity, the average salary for science and engineering faculty will eventually be greater than the average salary for arts and humanities faculty. $\boxed{As , x , approaches , positive , infinity, , the , average , salary , for , science , and , engineering , faculty , will , eventually , be , greater , than , the , average , salary , for , arts , and , humanities , faculty.}
Explanation
Understanding the Problem We are given the function g ( x ) = 15 3 x − 4 + 40 , which models the average salary (in thousands of dollars) for science and engineering faculty, where x is the number of years since the manager began recording the data. We want to determine which statement is true about the salaries of faculty members as x increases.
Analyzing the Function As x approaches positive infinity, we need to analyze the behavior of g ( x ) . The function g ( x ) is an increasing function because as x increases, the term 3 x − 4 also increases. Therefore, 15 3 x − 4 + 40 increases as well.
Comparing with Arts and Humanities Faculty Now, let's consider the possible scenarios for the average salary of arts and humanities faculty. It could be a constant, a linear function, a logarithmic function, or any other function. However, since g ( x ) increases without bound as x approaches infinity, the average salary for science and engineering faculty will eventually be greater than the average salary for arts and humanities faculty, regardless of how the average salary for arts and humanities faculty behaves.
Conclusion Therefore, the correct statement is: As x approaches positive infinity, the average salary for science and engineering faculty will eventually be greater than the average salary for arts and humanities faculty.
Examples
Imagine you're tracking the growth of two different investments over time. One investment, representing science and engineering faculty salaries, grows according to the function g ( x ) = 15 3 x − 4 + 40 . The other investment represents arts and humanities faculty salaries. Even if the arts and humanities investment starts higher or grows at a different rate initially, because the cube root function in g ( x ) continues to increase indefinitely (albeit slowly), eventually the science and engineering investment will surpass the arts and humanities investment. This illustrates how understanding the long-term behavior of functions can help predict future outcomes in various real-world scenarios, such as investment growth, population trends, or resource consumption.
The function g ( x ) = 15 3 x − 4 + 40 describes the average salary for science and engineering faculty, which increases without limit as x (years since recording began) increases. This means that eventually, the average salary for science and engineering faculty will surpass that of arts and humanities faculty. Therefore, the correct answer is option B.
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