For ` x in R , x ne0, 1, ` let `f_(0)(x)=(1)/(1-x) and f_(n+1)(x)=f_(0)(f_(n)(x)),n=0,1,2…..` Then the value of `f_(100)(3)+f_(1)((2)/(3))+f_(2)((3)/(2))` is equal to

Options

A. `(4)/(3)`
B. `(1)/(3)`
C. `(5)/(3)`
D. `(8)/(3)`

Explanation

To solve the problem, we need to find the value of \( f_{100}(3) + f_1\left(\frac{2}{3}\right) + f_2\left(\frac{3}{2}\right) \).

### Step 1: Define the functions

We start with the function definitions:
- \( f_0(x) = \frac{1}{1 - x} \)
- \( f_{n+1}(x) = f_0(f_n(x)) \)

### Step 2: Calculate \( f_1(x) \)

Using the definition of \( f_1(x) \):
\[ f_1(x) = f_0(f_0(x)) = f_0\left(\frac{1}{1 - x}\right) \]
Substituting into \( f_0 \):
\[ f_1(x) = f_0\left(\frac{1}{1 - x}\right) = \frac{1}{1 - \frac{1}{1 - x}} = \frac{1 - x}{-x} = \frac{x - 1}{x} \]

### Step 3: Calculate \( f_2(x) \)

Now, we calculate \( f_2(x) \):
\[ f_2(x) = f_0(f_1(x)) = f_0\left(\frac{x - 1}{x}\right) \]
Substituting into \( f_0 \):
\[ f_2(x) = f_0\left(\frac{x - 1}{x}\right) = \frac{1}{1 - \frac{x - 1}{x}} = \frac{1}{\frac{1}{x}} = x \]

### Step 4: Calculate \( f_3(x) \)

Next, we calculate \( f_3(x) \):
\[ f_3(x) = f_0(f_2(x)) = f_0(x) \]
Since \( f_2(x) = x \):
\[ f_3(x) = f_0(x) = \frac{1}{1 - x} \]

### Step 5: Identify the pattern

From the calculations, we can see a pattern:
- \( f_0(x) = \frac{1}{1 - x} \)
- \( f_1(x) = \frac{x - 1}{x} \)
- \( f_2(x) = x \)
- \( f_3(x) = \frac{1}{1 - x} \)

This pattern repeats every three iterations:
- \( f_{3n}(x) = f_0(x) \)
- \( f_{3n+1}(x) = f_1(x) \)
- \( f_{3n+2}(x) = f_2(x) \)

### Step 6: Calculate \( f_{100}(3) \)

Since \( 100 \mod 3 = 1 \):
\[ f_{100}(3) = f_1(3) = \frac{3 - 1}{3} = \frac{2}{3} \]

### Step 7: Calculate \( f_1\left(\frac{2}{3}\right) \)

Using the formula for \( f_1(x) \):
\[ f_1\left(\frac{2}{3}\right) = \frac{\frac{2}{3} - 1}{\frac{2}{3}} = \frac{-\frac{1}{3}}{\frac{2}{3}} = -\frac{1}{2} \]

### Step 8: Calculate \( f_2\left(\frac{3}{2}\right) \)

Since \( f_2(x) = x \):
\[ f_2\left(\frac{3}{2}\right) = \frac{3}{2} \]

### Step 9: Combine the results

Now we can combine all the results:
\[ f_{100}(3) + f_1\left(\frac{2}{3}\right) + f_2\left(\frac{3}{2}\right) = \frac{2}{3} - \frac{1}{2} + \frac{3}{2} \]

### Step 10: Simplify the expression

Finding a common denominator (which is 6):
\[ = \frac{4}{6} - \frac{3}{6} + \frac{9}{6} = \frac{4 - 3 + 9}{6} = \frac{10}{6} = \frac{5}{3} \]

### Final Answer

The value of \( f_{100}(3) + f_1\left(\frac{2}{3}\right) + f_2\left(\frac{3}{2}\right) \) is \( \frac{5}{3} \).