If `f(x)` = `1//(1-x)` then f f {f (x)}]=

Options

A. 0
B. x
C. `-x`
D. none

Answer

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

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

1. Start with the function:
\[ f(x) = \frac{1}{1-x} \]

2. Now, we need to find \( f(f(x)) \):
\[ f(f(x)) = f\left(\frac{1}{1-x}\right) \]

3. Substitute \( \frac{1}{1-x} \) into the function:
\[ f\left(\frac{1}{1-x}\right) = \frac{1}{1 - \frac{1}{1-x}} \]

4. Simplify the expression inside the function:
\[ 1 - \frac{1}{1-x} = \frac{(1-x) - 1}{1-x} = \frac{-x}{1-x} \]

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 \( f(f(f(x))) \):
\[ f(f(f(x))) = f\left(\frac{x-1}{x}\right) \]

2. Substitute \( \frac{x-1}{x} \) into the function:
\[ f\left(\frac{x-1}{x}\right) = \frac{1}{1 - \frac{x-1}{x}} \]

3. Simplify the expression inside the function:
\[ 1 - \frac{x-1}{x} = \frac{x - (x-1)}{x} = \frac{1}{x} \]

4. Therefore, we have:
\[ f(f(f(x))) = \frac{1}{\frac{1}{x}} = x \]

### Final Result

Thus, we find that:
\[ f(f(f(x))) = x \]