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The problem presents two exponential functions, \( f \) and \( g \), and asks for a comparison based on the following information:

1. **Function \( f \)** is represented in the table:
   - \( f(0) = 15 \)
   - \( f(1) = 7 \)
   - \( f(2) = 3 \)
   - \( f(3) = 1 \)
   - \( f(4) = 0 \)

2. **Function \( g \)** is an exponential function passing through two points: \( (0, 9) \) and \( (3, 0) \).

We need to find the correct statement comparing the behavior of both functions on the interval \( [0, 3] \).

### Key Observations:

- **Function \( f \)** is decreasing from \( x = 0 \) to \( x = 3 \) based on the table values.
- The second function, \( g \), passes through \( (0, 9) \) and \( (3, 0) \), indicating that it is also decreasing on the interval \( [0, 3] \).

### Rate of Decrease:
- Function \( f \) seems to decrease by about half each time:
  - \( f(0) = 15 \)
  - \( f(1) = 7 \) (about half of 15)
  - \( f(2) = 3 \) (about half of 7)
  - \( f(3) = 1 \) (about a third of 3)
  - \( f(4) = 0 \)

- Function \( g \) decreases from 9 to 0 over three units, indicating that it decreases continuously and eventually reaches zero.

Now let's analyze the answer options:

- **Option A**: Says both functions are decreasing on \( [0, 3] \), and function \( g \) is decreasing at a faster rate.
- **Option B**: Says only function \( f \) is decreasing on \( [0, 3] \), which is false since \( g \) is also decreasing.
- **Option C**: Says both functions are decreasing on \( [0, 3] \), and function \( f \) is decreasing at a faster rate.
- **Option D**: Says only function \( g \) is decreasing, which is false since \( f \) is also decreasing.

### Conclusion:
Both functions are decreasing, but function \( f \) seems to decrease more quickly since it reaches 1 by \( x = 3 \), while function \( g \) only approaches 0 at \( x = 3 \).

Thus, the correct answer is **Option C**.

Would you like a further explanation or details?

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### Additional Questions:
1. How do you calculate the rate of decrease for exponential functions?
2. How do you interpret the behavior of exponential functions graphically?
3. What are the key differences between exponential growth and decay?
4. How do you determine the exact function rule for exponential functions given points?
5. What is the significance of the function approaching zero in the context of exponential decay?

### Tip:
In exponential decay, a larger base in the exponential function leads to a faster rate of decay!

Select the correct answer.

Exponential function f is represented by the table:

x    0   1   2   3   4
f(x) 15  7   3   1   0

Function g is an exponential function passing through the points (0,9) and (3,0).

Which statement correctly compares the two functions?

A. Both functions are decreasing on [0, 3], and function g is decreasing at a faster rate.
B. Only function f is decreasing on [0, 3], and both functions are positive on that interval.
C. Both functions are decreasing on [0, 3], and function f is decreasing at a faster rate.
D. Only function g is decreasing on [0, 3], and only function f is positive on that interval.

Compare two exponential functions, f and g, where f is represented by a table and g passes through specific points. Analyze their rates of decrease on the interval [0, 3] to determine the correct comparison.

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