Given the function `f(x) = 1/(1-x)`. The points of discontinuity of the composite function, `y=f(f[f(x))]` are at x=

Options

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

Answer

To find the points of discontinuity of the composite function \( y = f(f(f(x))) \) where \( f(x) = \frac{1}{1 - x} \), we will follow these steps:

### Step 1: Understand the function \( f(x) \)
The function \( f(x) = \frac{1}{1 - x} \) is defined for all \( x \) except where the denominator is zero. Thus, \( f(x) \) is discontinuous when:
\[ 1 - x = 0 \implies x = 1 \]
So, \( f(x) \) is discontinuous at \( x = 1 \).

### Step 2: Find \( f(f(x)) \)
Next, we compute \( f(f(x)) \):
\[ f(f(x)) = f\left(\frac{1}{1 - x}\right) = \frac{1}{1 - \frac{1}{1 - x}} \]
Simplifying this, we have:
\[ = \frac{1}{\frac{(1 - x) - 1}{1 - x}} = \frac{1 - x}{-x} = \frac{x - 1}{x} \]
This function is defined for all \( x \) except where \( x = 0 \) (the denominator cannot be zero) and where \( f(x) \) is discontinuous, which we already found at \( x = 1 \).

### Step 3: Find \( f(f(f(x))) \)
Now, we compute \( f(f(f(x))) \):
\[ f(f(f(x))) = f\left(\frac{x - 1}{x}\right) = \frac{1}{1 - \frac{x - 1}{x}} \]
Simplifying this gives:
\[ = \frac{1}{\frac{x - (x - 1)}{x}} = \frac{1}{\frac{1}{x}} = x \]
This function is defined for all \( x \) except where the input to \( f \) is discontinuous. We already know \( f(x) \) is discontinuous at \( x = 1 \) and \( f(f(x)) \) is discontinuous at \( x = 0 \) and \( x = 1 \).

### Step 4: Identify points of discontinuity
Thus, we need to check the points of discontinuity:
1. **At \( x = 0 \)**: \( f(f(x)) \) is not defined.
2. **At \( x = 1 \)**: \( f(x) \) is not defined.

### Conclusion
The points of discontinuity of the composite function \( y = f(f(f(x))) \) are:
\[ \text{At } x = 0 \text{ and } x = 1. \]