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The question asks which function, \( f(x) \) or \( g(x) \), has a larger rate of growth. 

### Let's analyze each function:

#### Function \( g(x) = 2^x - 2 \):
This is an exponential function, and we know exponential functions grow rapidly. The base of the exponent is \( 2 \), which means the function grows by a factor of 2 as \( x \) increases.

#### Function \( f(x) \):
From the table:

| \( x \)  | \( f(x) \) |
|----------|------------|
| \( -2 \) | \( \frac{1}{9} \) |
| \( -1 \) | \( \frac{1}{3} \) |
| \( 0 \)  | \( 1 \)          |
| \( 1 \)  | \( 3 \)          |
| \( 2 \)  | \( 9 \)          |

Looking at the values of \( f(x) \), it appears that \( f(x) = 3^x \). This is also an exponential function, with base \( 3 \), which grows faster than base \( 2 \).

### Comparison:
Since \( f(x) = 3^x \) has a base of 3 and \( g(x) = 2^x - 2 \) has a base of 2, \( f(x) \) grows faster than \( g(x) \). Exponential functions with higher bases grow faster.

### Conclusion:
The function \( f(x) \) has a larger rate of growth.

Would you like more details on exponential growth, or do you have any other questions?

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Here are five related questions to deepen your understanding:
1. How would the growth rate compare if \( f(x) \) were \( 2^x \) instead of \( 3^x \)?
2. Can you explain why exponential functions with higher bases grow faster?
3. How would you find the exact intersection point of \( f(x) \) and \( g(x) \)?
4. What role does the constant term in \( g(x) = 2^x - 2 \) play in the function's growth?
5. How would the graph of \( f(x) = 3^x \) differ from \( g(x) = 2^x - 2 \)?

**Tip**: The base of an exponential function directly affects how fast it grows—the larger the base, the faster the growth.

Which function, f(x) or g(x), has a larger rate of growth? The table of f(x) values and the function g(x) = 2^x - 2 are given.

This problem involves comparing the growth rates of two exponential functions, f(x) = 3^x and g(x) = 2^x - 2. Explore how exponential growth rates differ based on the base of the exponent and determine which function grows faster. Suitable for students in Grades 9-11.

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