- Compare the functions at \(x=0\): \(q(0) = 1\) and \(r(0) = 125\).
- Analyze growth rates: The exponential function \(q(x) = (1.05)^x\) grows much faster than the linear function \(r(x) = 38x + 125\).
- Observe that the functions intersect at approximately \(x \approx 181.53\).
- Conclude that \(q(x) > r(x)\) only for very large values of \(x\), which corresponds to option (C).
- Final Answer: \(\boxed{\text{(C)}}\)
Explanation
- Initial Analysis of the Functions
We are comparing two functions defined for \(x > 0\): the exponential function \(q(x) = (1.05)^x\) and the linear function \(r(x) = 38x + 125\).
To understand their behavior, we first evaluate them at \(x = 0\):
- For the exponential function: \(q(0) = (1.05)^0 = 1\).
- For the linear function: \(r(0) = 38(0) + 125 = 125\).
At \(x=0\), the linear function is significantly larger than the exponential function (\(125 > 1\)).
- Comparison of Growth Rates
An exponential function with a base greater than 1, such as \(q(x) = (1.05)^x\), grows faster than any linear function \(r(x) = mx + c\) as \(x\) increases towards infinity.
Although the linear function \(r(x)\) has a much larger starting value, the exponential function's rate of growth will eventually allow it to catch up and surpass the linear function. This means there will be an intersection point \(x^*\) such that for all \(x > x^*\), \(q(x) > r(x)\).
- Determining the Relationship
To confirm this, we look for the point where \(q(x) = r(x)\). Setting up the equation \((1.05)^x = 38x + 125\), we find that the functions intersect at approximately \(x \approx 181.53\).
For any value of \(x\) smaller than this intersection point, the linear function \(r(x)\) is greater than \(q(x)\). For any value of \(x\) greater than this point, the exponential function \(q(x)\) is greater than \(r(x)\). Therefore, it is correct to say that the exponential function eventually exceeds the linear function. This matches the description in option (C).
- Conclusion
Since the exponential function \(q(x)\) is initially smaller than the linear function \(r(x)\) but grows much faster, it surpasses the linear function only for very large values of \(x\). Comparing this to the given options, we conclude that (C) is the true statement. The final answer is (C).
Examples
The relationship between exponential and linear growth is a fundamental concept in finance and population studies. For instance, consider a bank account earning compound interest versus a simple savings plan that adds a fixed amount every year. Initially, a large initial investment in the savings plan might outperform a small account earning interest. However, over a long enough period, the power of compounding (exponential growth) will inevitably cause the interest-bearing account to grow much larger than the savings account, demonstrating how initial conditions and growth models determine long-term outcomes.