Let `f(x) and g(x)`be functions which take integers as arguments. Let `f(x + y) =f(x)+ g(y) + 8` for all intege x and y. Let f(x) = x for all negative integers x and let `g (8) = 17`. Find f(0).

Answer

To solve the problem step by step, we start with the given functional equation and the conditions provided.

### Step 1: Understand the functional equation
We have the equation:
\[ f(x + y) = f(x) + g(y) + 8 \]
This holds for all integers \( x \) and \( y \).

### Step 2: Substitute \( y = 8 \)
We know that \( g(8) = 17 \). Let's substitute \( y = 8 \) into the functional equation:
\[ f(x + 8) = f(x) + g(8) + 8 \]
Substituting \( g(8) = 17 \) gives us:
\[ f(x + 8) = f(x) + 17 + 8 \]
This simplifies to:
\[ f(x + 8) = f(x) + 25 \]

### Step 3: Substitute \( x = -8 \)
Now, we want to find \( f(0) \). We can set \( x = -8 \) in the equation we derived:
\[ f(-8 + 8) = f(-8) + 25 \]
This simplifies to:
\[ f(0) = f(-8) + 25 \]

### Step 4: Find \( f(-8) \)
According to the problem, we have that \( f(x) = x \) for all negative integers \( x \). Since \(-8\) is a negative integer:
\[ f(-8) = -8 \]

### Step 5: Substitute \( f(-8) \) back into the equation for \( f(0) \)
Now we can substitute \( f(-8) = -8 \) into the equation we found for \( f(0) \):
\[ f(0) = -8 + 25 \]
This simplifies to:
\[ f(0) = 17 \]

### Conclusion
Thus, the final answer is:
\[ f(0) = 17 \]