`f(x)=(1)/(1-x)and f^(n)=fofof....of`, then the points of discontinuitym of f^(3n)(x) is/are

Options

A. x=2
B. x=0,1
C. x=1,2
D. none of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = \frac{1}{1-x} \) and determine the points of discontinuity for the function \( f^{(3n)}(x) \), which represents the function \( f \) composed with itself \( 3n \) times.

### Step 1: Identify the points of discontinuity of \( f(x) \)

The function \( f(x) = \frac{1}{1-x} \) is defined for all \( x \) except where the denominator is zero. Therefore, we set the denominator equal to zero to find the points of discontinuity:

\[ 1 - x = 0 \implies x = 1 \]

Thus, \( f(x) \) is discontinuous at \( x = 1 \).

### Step 2: Determine the behavior of \( f(x) \)

Next, we analyze the behavior of \( f(x) \) as \( x \) approaches the point of discontinuity:

- As \( x \) approaches \( 1 \) from the left (\( x \to 1^- \)), \( f(x) \to +\infty \).
- As \( x \) approaches \( 1 \) from the right (\( x \to 1^+ \)), \( f(x) \to -\infty \).

This confirms that \( x = 1 \) is indeed a point of discontinuity.

### Step 3: Find \( f^{(2)}(x) \)

Now, we will find \( f^{(2)}(x) = f(f(x)) \):

\[ f^{(2)}(x) = f\left(f(x)\right) = f\left(\frac{1}{1-x}\right) \]

Calculating \( f\left(\frac{1}{1-x}\right) \):

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

### Step 4: Identify points of discontinuity of \( f^{(2)}(x) \)

Now, we need to find the points of discontinuity of \( f^{(2)}(x) = \frac{x-1}{x} \):

The function \( \frac{x-1}{x} \) is discontinuous where the denominator is zero:

\[ x = 0 \]

Thus, \( f^{(2)}(x) \) is discontinuous at \( x = 0 \).

### Step 5: Find \( f^{(3)}(x) \)

Next, we find \( f^{(3)}(x) = f(f^{(2)}(x)) = f\left(\frac{x-1}{x}\right) \):

Calculating \( f\left(\frac{x-1}{x}\right) \):

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

### Step 6: Identify points of discontinuity of \( f^{(3)}(x) \)

The function \( f^{(3)}(x) = x \) is continuous for all \( x \). Therefore, it does not introduce any new points of discontinuity.

### Step 7: Generalize to \( f^{(3n)}(x) \)

Since \( f^{(3)}(x) = x \) is continuous, we can conclude that \( f^{(3n)}(x) \) for any integer \( n \) will also be continuous.

### Conclusion

The only point of discontinuity for \( f^{(3n)}(x) \) is at \( x = 1 \) from the first iteration of \( f(x) \).

Thus, the points of discontinuity of \( f^{(3n)}(x) \) are:

\[ \text{Points of discontinuity: } x = 1 \]