If `f(x)=(x+1)/(x-1), x ne 1`, find `f(f(f(f(2)))))`

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem of finding \( f(f(f(f(2)))) \) for the function \( f(x) = \frac{x+1}{x-1} \), we will follow these steps:

### Step 1: Calculate \( f(2) \)
Using the function definition:
\[ f(2) = \frac{2 + 1}{2 - 1} = \frac{3}{1} = 3 \]

### Step 2: Calculate \( f(f(2)) = f(3) \)
Now we need to find \( f(3) \):
\[ f(3) = \frac{3 + 1}{3 - 1} = \frac{4}{2} = 2 \]

### Step 3: Calculate \( f(f(f(2))) = f(f(3)) = f(2) \)
Next, we find \( f(f(3)) \), which we already calculated as \( f(2) \):
\[ f(f(3)) = f(2) = 3 \]

### Step 4: Calculate \( f(f(f(f(2)))) = f(f(f(3))) = f(3) \)
Finally, we find \( f(f(f(2))) \):
\[ f(f(f(2))) = f(3) = 2 \]

### Conclusion
Thus, the value of \( f(f(f(f(2)))) \) is \( 2 \).

### Summary of Steps:
1. Calculate \( f(2) = 3 \).
2. Calculate \( f(3) = 2 \).
3. Calculate \( f(f(2)) = f(3) = 2 \).
4. Calculate \( f(f(f(2))) = f(2) = 3 \).
5. Calculate \( f(f(f(f(2)))) = f(3) = 2 \).