To solve the problem, we need to analyze the function \( f \) defined on the set of integers \( \mathbb{Z} \) with the given conditions:
1. \( f(0) = 1 \)
2. \( f(f(n)) = f(f(n + 2)) + 2 \)
We need to find the values of \( f(3) \), \( f(2) \), and \( f(-2) \), and determine if \( f \) is a many-one function.
### Step 1: Analyze the function using the given conditions
We start with the first condition:
- \( f(0) = 1 \)
Now, we will use the second condition:
- \( f(f(n)) = f(f(n + 2)) + 2 \)
### Step 2: Substitute values into the second condition
Let’s substitute \( n = 0 \) into the second condition:
- \( f(f(0)) = f(f(2)) + 2 \)
Using \( f(0) = 1 \):
- \( f(1) = f(f(2)) + 2 \) (1)
### Step 3: Substitute \( n = 2 \)
Now, substitute \( n = 2 \) into the second condition:
- \( f(f(2)) = f(f(4)) + 2 \) (2)
### Step 4: Substitute \( n = 1 \)
Next, substitute \( n = 1 \):
- \( f(f(1)) = f(f(3)) + 2 \) (3)
### Step 5: Substitute \( n = 3 \)
Now, substitute \( n = 3 \):
- \( f(f(3)) = f(f(5)) + 2 \) (4)
### Step 6: Analyze the equations
From equations (1), (2), (3), and (4), we can see that we have a recursive relationship. We need to find specific values for \( f(1) \), \( f(2) \), \( f(3) \), and \( f(4) \).
### Step 7: Assume values and check consistency
Let’s assume \( f(2) = 0 \). Then from (1):
- \( f(1) = f(0) + 2 = 1 + 2 = 3 \)
Now substitute \( f(2) = 0 \) into (2):
- \( f(0) = f(f(4)) + 2 \)
- \( 1 = f(f(4)) + 2 \)
- Therefore, \( f(f(4)) = -1 \)
Now substitute \( f(1) = 3 \) into (3):
- \( f(3) = f(f(3)) + 2 \)
### Step 8: Check for \( f(3) \)
Let’s assume \( f(3) = -2 \) and check:
- Then from (4), we have \( f(-2) = f(f(5)) + 2 \).
### Step 9: Check if \( f \) is many-one
To determine if \( f \) is many-one, we need to check if there are distinct integers that map to the same value.
### Conclusion
After analyzing the equations and substituting values, we find:
- \( f(3) = -2 \)
- \( f(2) = 0 \)
- \( f(1) = 3 \)
Thus, the answers are:
- \( f(3) = -2 \) (Option c)
- \( f(2) = 0 \) (Option b)
