Let `f(x)=(1)/(1-x)."Then (fo(fof)) (x)"`

Options

A. x for all ` x in R`
B. x for all `x in R-{1}`
C. x for all `x in R-{0,1}`
D. None of these

Explanation

To solve the problem step by step, we need to find \( f(f(f(x))) \) where \( f(x) = \frac{1}{1 - x} \).

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

1. Start with the function \( f(x) = \frac{1}{1 - x} \).
2. Substitute \( f(x) \) into itself:
\[ f(f(x)) = f\left(\frac{1}{1 - x}\right) \]
3. Now, replace \( x \) in \( f(x) \) with \( \frac{1}{1 - x} \):
\[ f(f(x)) = \frac{1}{1 - \frac{1}{1 - x}} \]
4. Simplify the expression:
\[ 1 - \frac{1}{1 - x} = \frac{(1 - x) - 1}{1 - x} = \frac{-x}{1 - x} \]
5. Therefore, we have:
\[ f(f(x)) = \frac{1}{\frac{-x}{1 - x}} = \frac{1 - x}{-x} = \frac{x - 1}{x} \]

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

1. Now we need to find \( f(f(f(x))) = f\left(f(f(x))\right) = f\left(\frac{x - 1}{x}\right) \).
2. Substitute \( \frac{x - 1}{x} \) into \( f(x) \):
\[ f(f(f(x))) = f\left(\frac{x - 1}{x}\right) = \frac{1}{1 - \frac{x - 1}{x}} \]
3. Simplify the expression:
\[ 1 - \frac{x - 1}{x} = \frac{x - (x - 1)}{x} = \frac{1}{x} \]
4. Therefore, we have:
\[ f(f(f(x))) = \frac{1}{\frac{1}{x}} = x \]

### Final Result

Thus, the final result is:
\[ f(f(f(x))) = x \]
for \( x \) not equal to \( 0 \) and \( 1 \).