`f(x)=(1-x)/(1+x),x=-1` then `f^(-1)(x)` relation to

Options

A. f(x)
B. `(1)/(f(x))`
C. `-f(x)`
D. `-(1)/(f(x))`

Explanation

To solve the problem, we need to find the relationship between the function \( f(x) = \frac{1 - x}{1 + x} \) and its inverse \( f^{-1}(x) \).

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = \frac{1 - x}{1 + x} \]

2. **Set \( y = f(x) \)**:
\[ y = \frac{1 - x}{1 + x} \]

3. **Express \( x \) in terms of \( y \)**:
We need to solve for \( x \) in terms of \( y \). Start by cross-multiplying:
\[ y(1 + x) = 1 - x \]

4. **Distribute \( y \)**:
\[ y + yx = 1 - x \]

5. **Rearrange the equation**:
Move all terms involving \( x \) to one side:
\[ yx + x = 1 - y \]

6. **Factor out \( x \)**:
\[ x(y + 1) = 1 - y \]

7. **Solve for \( x \)**:
\[ x = \frac{1 - y}{y + 1} \]

8. **Identify the inverse function**:
Since we set \( y = f(x) \), we have:
\[ f^{-1}(y) = \frac{1 - y}{y + 1} \]

9. **Substituting back \( y \) with \( x \)**:
Thus, we can write:
\[ f^{-1}(x) = \frac{1 - x}{1 + x} \]

10. **Compare \( f^{-1}(x) \) with \( f(x) \)**:
We see that:
\[ f^{-1}(x) = f(x) \]

### Conclusion:
The relationship between \( f^{-1}(x) \) and \( f(x) \) is:
\[ f^{-1}(x) = f(x) \]