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Let's solve the problem step by step.

### Problem Statement Recap:
1. The function \( f(x) \) is defined by:
   \[
   f(x) = \frac{1 - x}{1 + x}, \quad x \neq -1.
   \]
   
2. **First part**: Show that \( f(f(x)) = x \).

3. **Second part**: Write down \( f^{-1}(x) \), the inverse of \( f(x) \).

### Solution:

#### Part 1: Showing that \( f(f(x)) = x \)

Given \( f(x) = \frac{1-x}{1+x} \), let's compute \( f(f(x)) \):

1. Substitute \( f(x) \) into itself:
   \[
   f(f(x)) = f\left(\frac{1-x}{1+x}\right)
   \]
2. Now, replace \( x \) in the original function with \( \frac{1-x}{1+x} \):
   \[
   f\left(\frac{1-x}{1+x}\right) = \frac{1 - \frac{1-x}{1+x}}{1 + \frac{1-x}{1+x}}
   \]

3. Simplify the expression step by step:
   \[
   1 - \frac{1-x}{1+x} = \frac{(1+x) - (1-x)}{1+x} = \frac{1+x - 1 + x}{1+x} = \frac{2x}{1+x}
   \]
   \[
   1 + \frac{1-x}{1+x} = \frac{(1+x) + (1-x)}{1+x} = \frac{1+x + 1-x}{1+x} = \frac{2}{1+x}
   \]

   So, the expression for \( f(f(x)) \) becomes:
   \[
   f(f(x)) = \frac{\frac{2x}{1+x}}{\frac{2}{1+x}} = x
   \]

Thus, we have shown that \( f(f(x)) = x \).

#### Part 2: Finding the Inverse \( f^{-1}(x) \)

To find the inverse function \( f^{-1}(x) \), we need to solve the equation \( y = f(x) \) for \( x \):

1. Start with:
   \[
   y = \frac{1 - x}{1 + x}
   \]
2. Cross-multiply to clear the fraction:
   \[
   y(1 + x) = 1 - x
   \]
   \[
   y + yx = 1 - x
   \]

3. Collect terms involving \( x \) on one side:
   \[
   yx + x = 1 - y
   \]
   \[
   x(y + 1) = 1 - y
   \]

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

So, the inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = \frac{1 - x}{1 + x}
\]

### Summary:
- **First part**: \( f(f(x)) = x \).
- **Second part**: The inverse function \( f^{-1}(x) \) is \( f^{-1}(x) = \frac{1 - x}{1 + x} \).

Would you like further details or have any questions? 

### Related Questions:
1. How do you determine if a function is bijective?
2. What are the necessary steps to prove that a function is invertible?
3. Can \( f^{-1}(x) \) have the same form as \( f(x) \) for all functions?
4. How does the condition \( x \neq -1 \) affect the domain and range of \( f(x) \)?
5. What are the implications of a function having \( f(f(x)) = x \)?

### Tip:
When dealing with functions and their inverses, always check if the composition of the function with its supposed inverse yields the identity function \( x \).

The function f(x) is defined by f(x) = (1 - x) / (1 + x), x ≠ -1. Show that f(f(x)) = x. Hence, write down f^(-1)(x).

Learn how to prove that f(f(x)) = x for the function f(x) = (1 - x) / (1 + x), and find the inverse function f^(-1)(x). This problem covers advanced concepts in functions, including composition and inverse functions, suitable for high school students in Grades 11-12.

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