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The problem involves comparing a linear function \( f(x) = 75x + 300 \) and an exponential function \( g(x) = 300(1.1)^x \), based on the given table values. The goal is to determine the \(x\)-value at which the exponential function begins to show more growth than the linear function.

### Observations from the Table:

| \(x\) | \(75x + 300\) | \(300(1.1)^x\) |
|------|--------------|--------------|
| 0    | 300          | 300          |
| 3    | 525          | 399.3        |
| 10   | 1050         | 778.123      |
| 15   | 1425         | 1253.17      |
| 18   | 1650         | 1667.98      |

At \(x = 18\), the exponential function \( g(x) = 300(1.1)^x \) surpasses the linear function \( f(x) = 75x + 300 \) because \( 1667.98 > 1650 \). 

Thus, the exponential function shows more growth than the linear function at **\(x = 18\)**.

Would you like a detailed explanation of how this conclusion is derived? 

Here are five related questions to deepen understanding:
1. How can you identify the growth rate difference between linear and exponential functions?
2. Why does an exponential function eventually surpass a linear function in terms of growth?
3. Can you calculate when an exponential function overtakes a linear one for different growth rates?
4. How does changing the base of an exponential function affect when it surpasses a linear function?
5. What other types of functions grow faster than linear but slower than exponential?

**Tip**: Exponential functions grow at a constant percentage rate, while linear functions grow at a constant amount per unit increase in \(x\).

Addi has the two functions f(x) = 75x + 300 and g(x) = 300(1.1)^x. She wants to determine if the exponential function will ever show more growth than the linear function. At which x-value does the exponential function show more growth than the linear function?

Explore the comparison between a linear function f(x) = 75x + 300 and an exponential function g(x) = 300(1.1)^x. Learn when the exponential function surpasses the linear function, using a detailed table of values. Suitable for students in Grades 9-12.

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