**A: What type of function, linear or exponential, can be used to describe the value of the investment after a fixed number of years using option 1 and option 2? Explain your answer.**
Option 1: exponential
Option 2: linear
**Explanation:**
An exponential function is a function in which the output increases or decreases by a constant factor each time the input increases by a constant amount. In Option 1, the value of the investment increases by a factor of 1.3 each year. This is because the value of the investment is multiplied by 1.3 each year.
A linear function is a function in which the output increases or decreases by a constant amount each time the input increases by a constant amount. In Option 2, the value of the investment increases by $300 each year.
**B: Write one function for each option to describe the value of the investment f(n), in dollars, after n years.**
Option 1:
```
f(n) = 1000 * 1.3^n
```
Option 2:
```
f(n) = 1000 + 300n
```
**C: Belinda wants to invest in an option that would help to increase her investment value by the greatest amount in 20 years. Will there be any significant difference in the value of Belinda's investment 20 years if she uses option 2 over option 1? Explain your answer, and show the investment value after 20 years for each option.**
**Option 1:**
```
f(n) = 1000 * 1.3^n
```
Substituting n = 20 into the equation, we get:
```
f(20) = 1000 * 1.3^20 = $61,626.41
```
**Option 2:**
```
f(n) = 1000 + 300n
```
Substituting n = 20 into the equation, we get:
```
f(20) = 1000 + 300 * 20 = $7000
```
**Conclusion:**
Option 1 will result in a significantly higher investment value after 20 years than Option 2. The investment value using Option 1 will be over $54,000 greater than the investment value using Option 2.
This is because exponential functions grow much faster than linear functions over time. In this case, the investment value in Option 1 is multiplied by 1.3 each year, while the investment value in Option 2 only increases by $300 each year.
**Recommendation:**
Belinda should invest in Option 1 if she wants to maximize the growth of her investment over time.