`f_(n)={{:(f_(n-1)" if n is even"),(2f_(n-1)" if n is odd"):}`
and `f_(o)=1`, find the value of `f_(4)+f_(5)`

Answer

To solve the problem, we will follow the recursive definition of the function \( f(n) \) given in the question. The function is defined as follows:

- \( f(n) = f(n-1) \) if \( n \) is even
- \( f(n) = 2 \cdot f(n-1) \) if \( n \) is odd
- \( f(0) = 1 \)

We need to find the value of \( f(4) + f(5) \).

### Step-by-Step Solution:

1. **Calculate \( f(4) \)**:
- Since \( 4 \) is even, we use the first rule:
\[ f(4) = f(3) \]

2. **Calculate \( f(3) \)**:
- Since \( 3 \) is odd, we use the second rule:
\[ f(3) = 2 \cdot f(2) \]

3. **Calculate \( f(2) \)**:
- Since \( 2 \) is even, we use the first rule:
\[ f(2) = f(1) \]

4. **Calculate \( f(1) \)**:
- Since \( 1 \) is odd, we use the second rule:
\[ f(1) = 2 \cdot f(0) \]

5. **Calculate \( f(0) \)**:
- We know from the problem statement that:
\[ f(0) = 1 \]

6. **Substituting back to find \( f(1) \)**:
- Now substituting \( f(0) \) into the equation for \( f(1) \):
\[ f(1) = 2 \cdot 1 = 2 \]

7. **Substituting back to find \( f(2) \)**:
- Now substituting \( f(1) \) into the equation for \( f(2) \):
\[ f(2) = f(1) = 2 \]

8. **Substituting back to find \( f(3) \)**:
- Now substituting \( f(2) \) into the equation for \( f(3) \):
\[ f(3) = 2 \cdot f(2) = 2 \cdot 2 = 4 \]

9. **Substituting back to find \( f(4) \)**:
- Now substituting \( f(3) \) into the equation for \( f(4) \):
\[ f(4) = f(3) = 4 \]

10. **Calculate \( f(5) \)**:
- Since \( 5 \) is odd, we use the second rule:
\[ f(5) = 2 \cdot f(4) = 2 \cdot 4 = 8 \]

11. **Final Calculation**:
- Now we can find \( f(4) + f(5) \):
\[ f(4) + f(5) = 4 + 8 = 12 \]

Thus, the final answer is:
\[ \boxed{12} \]