Natalie buys a new car. At the end of the first month, the odometer on the car read...

1. Identify the initial condition: The problem states that at the end of the first month, the odometer reads 800 miles. This means $f(1) = 800$. This is the starting point of our function.

1. Identify the rate of change: Natalie expects to drive 900 miles per month. This is the constant amount added to the odometer each month after the first month. This indicates an arithmetic progression.

3. Formulate a recursive function: A recursive function defines a term in a sequence based on previous terms. Since 900 miles are added each month, the number of miles at month $n$ can be found by taking the number of miles at month $n-1$ and adding 900. This can be written as $f(n) = f(n-1) + 900$, for $n \ge 2$. We also need the initial condition $f(1) = 800$. Therefore, one correct representation is:
$$f(1) = 800$$
$$f(n) = f(n-1) + 900, \text{ for } n \ge 2$$ This matches one of the options.

4. Formulate an explicit function: An explicit function expresses the term directly in terms of $n$. For an arithmetic sequence, the formula is $a_n = a_1 + d(n-1)$, where $a_1$ is the first term and $d$ is the common difference. In this case, $a_1 = f(1) = 800$ and $d = 900$. So, the explicit function is:
$$f(n) = 800 + 900(n-1)$$ We can expand this: $f(n) = 800 + 900n - 900$, which simplifies to $f(n) = 900n - 100$. This matches another option.

5. Verify the explicit function: Let's check if $f(n) = 900n - 100$ gives the correct values for the first few months.
For $n=1$: $f(1) = 900(1) - 100 = 900 - 100 = 800$. This is correct.
For $n=2$: $f(2) = 900(2) - 100 = 1800 - 100 = 1700$. Using the recursive definition, $f(2) = f(1) + 900 = 800 + 900 = 1700$. This is also correct.

6. Evaluate other options: Let's examine the other provided options to confirm they are incorrect.
- $f(n) = f(n-1) + 800, \text{ for } n \ge 2$ with $f(1) = 900$: This implies starting at 900 and adding 800 each month, which contradicts the problem statement.
- $f(n) = 900n + 800$: For $n=1$, $f(1) = 900(1) + 800 = 1700$, which is incorrect.
- $f(n) = f(n-1) + 100, \text{ for } n \ge 2$ with $f(1) = 800$: This implies adding only 100 miles per month, which is incorrect.
- $f(n) = 900n - 100$: This is the correct explicit form derived in step 4.
- $f(n) = 800n + 100$: For $n=1$, $f(1) = 800(1) + 100 = 900$, which is incorrect.

Based on the analysis, the correct functions are the recursive definition and the explicit definition $f(n) = 900n - 100$. The explicit form $f(n) = 800 + 900(n-1)$ is also equivalent to $f(n) = 900n - 100$. The recursive form $f(1) = 800, f(n) = f(n-1) + 900, \text{ for } n \ge 2$ is also correct.