Question

Determine if the function defined in the table has a positive or negative rate of change:

X f(x)
1 5
2 10
3 20
4 40
5 70


x g(x)
1 15
2 25
3 33
4 38
5 40

Answer 1

To determine the rate of change of the functions provided in the tables, we can look at how the values of \( f(x) \) and \( g(x) \) change as \( x \) increases.

**For \( f(x) \):**
- When \( x = 1 \), \( f(x) = 5 \)
- When \( x = 2 \), \( f(x) = 10 \) (change: \( 10 - 5 = 5 \))
- When \( x = 3 \), \( f(x) = 20 \) (change: \( 20 - 10 = 10 \))
- When \( x = 4 \), \( f(x) = 40 \) (change: \( 40 - 20 = 20 \))
- When \( x = 5 \), \( f(x) = 70 \) (change: \( 70 - 40 = 30 \))

The changes in \( f(x) \) are as follows:
- From \( x = 1 \) to \( x = 2 \): \( 5 \)
- From \( x = 2 \) to \( x = 3 \): \( 10 \)
- From \( x = 3 \) to \( x = 4 \): \( 20 \)
- From \( x = 4 \) to \( x = 5 \): \( 30 \)

All changes are positive, indicating that the function \( f(x) \) has a **positive rate of change**.

---

**For \( g(x) \):**
- When \( x = 1 \), \( g(x) = 15 \)
- When \( x = 2 \), \( g(x) = 25 \) (change: \( 25 - 15 = 10 \))
- When \( x = 3 \), \( g(x) = 33 \) (change: \( 33 - 25 = 8 \))
- When \( x = 4 \), \( g(x) = 38 \) (change: \( 38 - 33 = 5 \))
- When \( x = 5 \), \( g(x) = 40 \) (change: \( 40 - 38 = 2 \))

The changes in \( g(x) \) are as follows:
- From \( x = 1 \) to \( x = 2 \): \( 10 \)
- From \( x = 2 \) to \( x = 3 \): \( 8 \)
- From \( x = 3 \) to \( x = 4 \): \( 5 \)
- From \( x = 4 \) to \( x = 5 \): \( 2 \)

All of these changes are also positive, indicating that the function \( g(x) \) has a **positive rate of change**, though it is decreasing at a slower pace.

---

In summary:
- \( f(x) \) has a **positive rate of change**.
- \( g(x) \) also has a **positive rate of change**.