`f(x)=1-h(x), g(x)=1-k(x), h(x)=f(x)+1`
`f(x)=g(x)+1, k(x)=j(x)+1`
Find the value of `(f(f(x))+k(g(x)))/(h(k(x))+f(j(x)))`:

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To solve the problem step by step, we will first rewrite the given functions and then substitute values as we derive them from the relationships provided.

### Given Functions:
1. \( f(x) = 1 - h(x) \) (1)
2. \( g(x) = 1 - k(x) \) (2)
3. \( h(x) = f(x) + 1 \) (3)
4. \( f(x) = g(x) + 1 \) (4)
5. \( k(x) = j(x) + 1 \) (5)

### Step 1: Find \( f(x) \)
From equation (1) and (3):
\[ f(x) = 1 - h(x) \]
Substituting \( h(x) \) from (3):
\[ f(x) = 1 - (f(x) + 1) \]
This simplifies to:
\[ f(x) = 1 - f(x) - 1 \]
\[ f(x) + f(x) = 0 \]
\[ 2f(x) = 0 \implies f(x) = 0 \]

### Step 2: Find \( h(x) \)
Using the value of \( f(x) \):
\[ h(x) = f(x) + 1 = 0 + 1 = 1 \]

### Step 3: Find \( g(x) \)
Using \( f(x) \) in equation (4):
\[ f(x) = g(x) + 1 \implies 0 = g(x) + 1 \]
\[ g(x) = -1 \]

### Step 4: Find \( k(x) \)
Using \( g(x) \) in equation (2):
\[ g(x) = 1 - k(x) \implies -1 = 1 - k(x) \]
\[ k(x) = 2 \]

### Step 5: Find \( j(x) \)
Using \( k(x) \) in equation (5):
\[ k(x) = j(x) + 1 \implies 2 = j(x) + 1 \]
\[ j(x) = 1 \]

### Step 6: Substitute values into the expression
We need to find:
\[ \frac{f(f(x)) + k(g(x))}{h(k(x)) + f(j(x))} \]

Calculating each part:
- \( f(f(x)) = f(0) = 0 \) (since \( f(x) = 0 \))
- \( g(x) = -1 \) so \( k(g(x)) = k(-1) = 2 \)
- Thus, \( f(f(x)) + k(g(x)) = 0 + 2 = 2 \)

Now for the denominator:
- \( h(k(x)) = h(2) = 1 \) (since \( h(x) = 1 \) for any \( x \))
- \( f(j(x)) = f(1) = 0 \)
- Thus, \( h(k(x)) + f(j(x)) = 1 + 0 = 1 \)

### Final Calculation:
\[ \frac{f(f(x)) + k(g(x))}{h(k(x)) + f(j(x))} = \frac{2}{1} = 2 \]

### Final Answer:
The value is \( 2 \).