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
`(h(f(k(x)))+j(g(h(x)))+k(j(f(x))))/[f(x)+j(x)+k(x)][f(x).j(x)k(x)]` is equal to :

Options

A. 0
B. 3
C. `3//2`
D. none of these

Answer

To solve the problem step by step, we will first analyze the given functions and their relationships.

1. **Given Functions**:
- \( f(x) = 1 - h(x) \)
- \( g(x) = 1 - k(x) \)
- \( h(x) = f(x) + 1 \)
- \( f(x) = g(x) + 1 \)
- \( k(x) = j(x) + 1 \)

2. **Substituting \( h(x) \) into \( f(x) \)**:
Start with the equation \( f(x) = 1 - h(x) \).
Substitute \( h(x) = f(x) + 1 \):
\[ f(x) = 1 - (f(x) + 1) \]
Simplifying this gives:
\[ f(x) = 1 - f(x) - 1 \]
\[ f(x) + f(x) = 0 \implies 2f(x) = 0 \implies f(x) = 0 \]

3. **Finding \( h(x) \)**:
Now that we have \( f(x) = 0 \), we can find \( h(x) \):
\[ h(x) = f(x) + 1 = 0 + 1 = 1 \]

4. **Finding \( g(x) \)**:
Using \( f(x) = g(x) + 1 \):
\[ 0 = g(x) + 1 \implies g(x) = -1 \]

5. **Finding \( k(x) \)**:
Using \( g(x) = 1 - k(x) \):
\[ -1 = 1 - k(x) \implies k(x) = 2 \]

6. **Finding \( j(x) \)**:
Using \( k(x) = j(x) + 1 \):
\[ 2 = j(x) + 1 \implies j(x) = 1 \]

7. **Summary of Values**:
- \( f(x) = 0 \)
- \( h(x) = 1 \)
- \( g(x) = -1 \)
- \( k(x) = 2 \)
- \( j(x) = 1 \)

8. **Calculating the Expression**:
We need to evaluate:
\[ \frac{h(f(k(x))) + j(g(h(x))) + k(j(f(x)))}{[f(x) + j(x) + k(x)][f(x) \cdot j(x) \cdot k(x)]} \]

- **Calculating \( h(f(k(x))) \)**:
Since \( k(x) = 2 \), we have \( f(k(x)) = f(2) = 0 \).
Thus, \( h(f(k(x))) = h(0) = 1 \).

- **Calculating \( j(g(h(x))) \)**:
Since \( h(x) = 1 \), we have \( g(h(x)) = g(1) = -1 \).
Thus, \( j(g(h(x))) = j(-1) = 1 \).

- **Calculating \( k(j(f(x))) \)**:
Since \( f(x) = 0 \), we have \( j(f(x)) = j(0) = 1 \).
Thus, \( k(j(f(x))) = k(1) = 2 \).

- **Putting it all together**:
The numerator becomes:
\[ 1 + 1 + 2 = 4 \]

- **Calculating the Denominator**:
\[ f(x) + j(x) + k(x) = 0 + 1 + 2 = 3 \]
\[ f(x) \cdot j(x) \cdot k(x) = 0 \cdot 1 \cdot 2 = 0 \]
Therefore, the denominator is:
\[ 3 \cdot 0 = 0 \]

9. **Final Expression**:
The expression becomes:
\[ \frac{4}{0} \]
This indicates that the expression is undefined.