If `f(x)={x ,xi sr a t ion a l1-x ,xi si r r a t ion a l ,t h e nf(f(x))` is `xAAx in R` (b) `{x , xi si r r a t ion a l1-x ,xi sr a t ion a l` `{x ,xi sr a t ion a l1-x ,xi si r r a t ion a l` (d) none of these

Options

A. `x AA x in R`
B. `f(x)={(x",",x" is irrational"),(1-x",",x" is rational"):}`
C. `f(x)={(x",",x" is rational"),(1-x",",x" is irrational"):}`
D. None of these

Explanation

To solve the problem, we need to analyze the function \( f(x) \) defined as follows:

\[ f(x) = \begin{cases} x & \text{if } x \text{ is rational} \\ 1 - x & \text{if } x \text{ is irrational} \end{cases} \]

We need to find \( f(f(x)) \) and determine its value based on the nature of \( x \) (whether it is rational or irrational).

### Step 1: Determine \( f(f(x)) \) when \( x \) is rational.

1. If \( x \) is rational, then by the definition of \( f \):
\[ f(x) = x \]
2. Now, we need to find \( f(f(x)) \):
\[ f(f(x)) = f(x) = f(x) = x \]

### Step 2: Determine \( f(f(x)) \) when \( x \) is irrational.

1. If \( x \) is irrational, then by the definition of \( f \):
\[ f(x) = 1 - x \]
2. Now, we need to find \( f(f(x)) \):
\[ f(f(x)) = f(1 - x) \]
3. We need to check whether \( 1 - x \) is rational or irrational:
- Since \( x \) is irrational, \( 1 - x \) is also irrational.
4. Therefore, we apply the second case of \( f \):
\[ f(1 - x) = 1 - (1 - x) = x \]

### Conclusion

In both cases (whether \( x \) is rational or irrational), we find that:
\[ f(f(x)) = x \]

Thus, the final answer is:
\[ f(f(x)) = x \quad \text{for all } x \in \mathbb{R} \]

### Answer
The correct option is (a) \( f(f(x)) = x \) for all \( x \in \mathbb{R} \).

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