Find `f(f(f(f(f(2))))))` if `f(x)=(x+1)/(x-1),x ne 1`

Options

A. 1
B. 2
C. 3
D. 5

Answer

To solve the problem of finding \( f(f(f(f(f(2))))) \) where \( f(x) = \frac{x+1}{x-1} \) and \( x \neq 1 \), we will proceed step by step.

### 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 \( f(f(3)) \):
\[ f(f(3)) = f(2) = 3 \quad \text{(from Step 1)} \]

### Step 4: Calculate \( f(f(f(f(2)))) = f(f(f(3))) = f(3) \)
Now we find \( f(f(f(3))) \):
\[ f(f(f(3))) = f(3) = 2 \quad \text{(from Step 2)} \]

### Step 5: Calculate \( f(f(f(f(f(2))))) = f(f(f(f(3)))) = f(2) \)
Finally, we find \( f(f(f(f(3)))) \):
\[ f(f(f(f(3)))) = f(2) = 3 \quad \text{(from Step 1)} \]

### Conclusion
Thus, the final result is:
\[ f(f(f(f(f(2))))) = 3 \]