If `f(x)=(x+1)/(x-1)` then the value of `f(f(f(x)))` is `:`

Options

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

Explanation

To find the value of \( f(f(f(x))) \) where \( f(x) = \frac{x+1}{x-1} \), we will follow these steps:

### Step 1: Find \( f(f(x)) \)

1. **Start with the function**:
\[ f(x) = \frac{x+1}{x-1} \]
2. **Substitute \( f(x) \) into itself**:
\[ f(f(x)) = f\left(\frac{x+1}{x-1}\right) \]
3. **Replace \( x \) in the function with \( \frac{x+1}{x-1} \)**:
\[ f\left(\frac{x+1}{x-1}\right) = \frac{\frac{x+1}{x-1} + 1}{\frac{x+1}{x-1} - 1} \]
4. **Simplify the numerator**:
\[ \frac{x+1}{x-1} + 1 = \frac{x+1 + (x-1)}{x-1} = \frac{2x}{x-1} \]
5. **Simplify the denominator**:
\[ \frac{x+1}{x-1} - 1 = \frac{x+1 - (x-1)}{x-1} = \frac{2}{x-1} \]
6. **Combine the results**:
\[ f(f(x)) = \frac{\frac{2x}{x-1}}{\frac{2}{x-1}} = x \]

### Step 2: Find \( f(f(f(x))) \)

1. **Now find \( f(f(f(x))) \)**:
\[ f(f(f(x))) = f(f(x)) \]
2. **Since we found that \( f(f(x)) = x \)**:
\[ f(f(f(x))) = f(x) \]
3. **Substituting back into the original function**:
\[ f(f(f(x))) = f(x) = \frac{x+1}{x-1} \]

### Final Result

Thus, the value of \( f(f(f(x))) \) is:
\[ \boxed{\frac{x+1}{x-1}} \]