Question

Determine if the function defined in the table has a positive or negative rate of change:
A. x: 1, 2, 3, 4, 5
f(x): 5, 10,20, 40, 70

B. x: 1, 2, 3, 4, 5
g(x): 15, 25, 33, 38, 40

Answer 1

To determine if the functions \( f(x) \) and \( g(x) \) have positive or negative rates of change, we can calculate the rate of change between consecutive points.

### For function \( f(x) \):
- \( f(1) = 5 \)
- \( f(2) = 10 \)
- \( f(3) = 20 \)
- \( f(4) = 40 \)
- \( f(5) = 70 \)

Now, let's compute the changes:
- \( f(2) - f(1) = 10 - 5 = 5 \)
- \( f(3) - f(2) = 20 - 10 = 10 \)
- \( f(4) - f(3) = 40 - 20 = 20 \)
- \( f(5) - f(4) = 70 - 40 = 30 \)

Since all these differences are positive, \( f(x) \) has a **positive rate of change**.

### For function \( g(x) \):
- \( g(1) = 15 \)
- \( g(2) = 25 \)
- \( g(3) = 33 \)
- \( g(4) = 38 \)
- \( g(5) = 40 \)

Now, let's compute the changes:
- \( g(2) - g(1) = 25 - 15 = 10 \)
- \( g(3) - g(2) = 33 - 25 = 8 \)
- \( g(4) - g(3) = 38 - 33 = 5 \)
- \( g(5) - g(4) = 40 - 38 = 2 \)

All differences are positive, so \( g(x) \) also has a **positive rate of change**.

### Conclusion
- The function \( f(x) \) has a **positive rate of change**.
- The function \( g(x) \) also has a **positive rate of change**.