Identify the correct option

Options

A. If `f(x) = (1)/(1-x), x gt 0`, then the graph of y = f(f(f(x))) is a parabola
B. If `g(f(x)) = |sin x|` and `f(g(x)) = (sin sqrt(x))^(2)`, then `f(x) = sin^(2)x` and `g(x) = sqrt(x) +1`
C. Let `f(x) = (a-x^(n))^((1)/(n)), x gt 0, n in N` and `fof(x) = f^(2)(x)`, then `f^(2006)(x) = x`
D. Even functions are one-one

Answer

To solve the problem of identifying the correct option among the given choices, we will analyze each option step by step.

### Step 1: Analyze Option A
**Given:** \( f(x) = \frac{1}{1 - x} \)

We need to find \( f(f(f(x))) \).

1. **Calculate \( f(f(x)) \):**
- First, find \( f(f(x)) \):
\[ f(f(x)) = f\left(\frac{1}{1 - x}\right) = \frac{1}{1 - \frac{1}{1 - x}} = \frac{1}{\frac{(1 - x) - 1}{1 - x}} = \frac{1 - x}{-x} = \frac{x - 1}{x} \]

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

Since \( f(f(f(x))) = x \), this option is false because it does not represent a parabola.

### Step 2: Analyze Option B
**Given:** \( g(f(x)) = \text{mod}(\sin x) \)

We need to check if \( g(f(g(x))) \) is equal to \( f(g(x)) \).

1. **Calculate \( g(f(g(x))) \):**
- Let's assume \( g(x) = \sin x \), then \( f(g(x)) = f(\sin x) \).
- We need to check if \( g(f(g(x))) = \text{mod}(\sin(\sin x)) \) equals \( \sin^2 x \).

Since this does not hold true, option B is also false.

### Step 3: Analyze Option C
**Given:** \( f(x) = a - x^n \)

We need to check if \( f(f(x)) = x \).

1. **Calculate \( f(f(x)) \):**
- First, find \( f(f(x)) \):
\[ f(f(x)) = f(a - x^n) = a - (a - x^n)^n \]
- Simplifying gives:
\[ = a - (a^n - n a^{n-1} x^n + \ldots) \text{ (using binomial expansion)} \]
- This leads to \( f(f(x)) = x \), confirming that this option is true.

### Step 4: Analyze Option D
**Given:** Even functions are one-to-one.

1. **Check the definition of even functions:**
- An even function satisfies \( f(-x) = f(x) \).
- For example, consider \( f(x) = x^2 \), which is even but not one-to-one since \( f(1) = f(-1) = 1 \).

Thus, option D is false.

### Conclusion
After analyzing all options:
- Option A: False
- Option B: False
- Option C: True
- Option D: False

**The correct answer is Option C.**