Given `f(x)=(1)/((1-x)).g(x)=f(f(x)) and h(x)=f(f(f(x)))`, then the value of `f(x).g(x).h(x)` is :

Options

A. 0
B. `-1`
C. 1
D. 2

Answer

To solve the problem, we need to find the values of \( g(x) \) and \( h(x) \) based on the function \( f(x) \) given as:

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

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

The function \( g(x) \) is defined as \( g(x) = f(f(x)) \).

First, we need to calculate \( f(f(x)) \):

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

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

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

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

So, we have:
\[ g(x) = \frac{x - 1}{x} \]

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

The function \( h(x) \) is defined as \( h(x) = f(f(f(x))) \).

1. We already know \( f(f(x)) = \frac{x - 1}{x} \), so we need to find \( f\left(g(x)\right) \):
\[ h(x) = f\left(\frac{x - 1}{x}\right) \]

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

So, we have:
\[ h(x) = x \]

### Step 3: Calculate \( f(x) \cdot g(x) \cdot h(x) \)

Now we can calculate \( f(x) \cdot g(x) \cdot h(x) \):

1. Substitute the values:
\[ f(x) = \frac{1}{1 - x}, \quad g(x) = \frac{x - 1}{x}, \quad h(x) = x \]

2. Multiply them together:
\[ f(x) \cdot g(x) \cdot h(x) = \left(\frac{1}{1 - x}\right) \cdot \left(\frac{x - 1}{x}\right) \cdot x \]

3. Simplifying:
\[ = \frac{1 \cdot (x - 1) \cdot x}{(1 - x) \cdot x} = \frac{x - 1}{1 - x} \]

4. Notice that \( \frac{x - 1}{1 - x} = -1 \).

Thus, the final value is:
\[ f(x) \cdot g(x) \cdot h(x) = -1 \]

### Final Answer:
The value of \( f(x) \cdot g(x) \cdot h(x) \) is \( -1 \).

---