Following questions are based on the given information for the following functions `f(x)`
`f(x)=2bx+f(-x), if x lt0`
`f(x)=a if x =0`
`f(x)=b+c-2cx+f(x-1), if x gt0`
`f(8)` equals :

Options

A. `a+8b-32c`
B. `a+8(b-8c)`
C. `8(a+b-c)`
D. none of these

Answer

To solve the problem, we need to evaluate the function \( f(x) \) based on the given definitions for different intervals of \( x \). The function is defined piecewise:

1. \( f(x) = 2bx + f(-x) \) for \( x < 0 \)
2. \( f(x) = a \) for \( x = 0 \)
3. \( f(x) = b + c - 2cx + f(x-1) \) for \( x > 0 \)

We need to find \( f(8) \).

### Step 1: Calculate \( f(1) \)
Using the third definition for \( x = 1 \):
\[ f(1) = b + c - 2c(1) + f(0) \]
Substituting \( f(0) = a \):
\[ f(1) = b + c - 2c + a = b - c + a \]

### Step 2: Calculate \( f(2) \)
Using the third definition for \( x = 2 \):
\[ f(2) = b + c - 2c(2) + f(1) \]
Substituting \( f(1) = b - c + a \):
\[ f(2) = b + c - 4c + (b - c + a) = 2b - 4c + a \]

### Step 3: Calculate \( f(3) \)
Using the third definition for \( x = 3 \):
\[ f(3) = b + c - 2c(3) + f(2) \]
Substituting \( f(2) = 2b - 4c + a \):
\[ f(3) = b + c - 6c + (2b - 4c + a) = 3b - 9c + a \]

### Step 4: Calculate \( f(4) \)
Using the third definition for \( x = 4 \):
\[ f(4) = b + c - 2c(4) + f(3) \]
Substituting \( f(3) = 3b - 9c + a \):
\[ f(4) = b + c - 8c + (3b - 9c + a) = 4b - 16c + a \]

### Step 5: Calculate \( f(5) \)
Using the third definition for \( x = 5 \):
\[ f(5) = b + c - 2c(5) + f(4) \]
Substituting \( f(4) = 4b - 16c + a \):
\[ f(5) = b + c - 10c + (4b - 16c + a) = 5b - 25c + a \]

### Step 6: Calculate \( f(6) \)
Using the third definition for \( x = 6 \):
\[ f(6) = b + c - 2c(6) + f(5) \]
Substituting \( f(5) = 5b - 25c + a \):
\[ f(6) = b + c - 12c + (5b - 25c + a) = 6b - 36c + a \]

### Step 7: Calculate \( f(7) \)
Using the third definition for \( x = 7 \):
\[ f(7) = b + c - 2c(7) + f(6) \]
Substituting \( f(6) = 6b - 36c + a \):
\[ f(7) = b + c - 14c + (6b - 36c + a) = 7b - 49c + a \]

### Step 8: Calculate \( f(8) \)
Using the third definition for \( x = 8 \):
\[ f(8) = b + c - 2c(8) + f(7) \]
Substituting \( f(7) = 7b - 49c + a \):
\[ f(8) = b + c - 16c + (7b - 49c + a) = 8b - 64c + a \]

### Final Result
Thus, we find that:
\[ f(8) = 8b - 64c + a \]