If `f(x)={{:(1,x" is a rational number"),(0",","x is an irrational number"):}"what is/are the"`
value (s) of `(fof)` `(sqrt3)` ?

Options

A. 0
B. 1
C. Both 0 and 1
D. None of these

Explanation

To solve the problem, we need to evaluate \( f(f(\sqrt{3})) \) given the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} 1 & \text{if } x \text{ is a rational number} \\ 0 & \text{if } x \text{ is an irrational number} \end{cases} \]

### Step-by-Step Solution:

**Step 1: Evaluate \( f(\sqrt{3}) \)**

- First, we need to determine whether \( \sqrt{3} \) is a rational or an irrational number.
- Since \( \sqrt{3} \) cannot be expressed as a fraction of two integers, it is classified as an **irrational number**.

Thus, according to the function definition:
\[ f(\sqrt{3}) = 0 \quad \text{(since } \sqrt{3} \text{ is irrational)} \]

**Step 2: Evaluate \( f(f(\sqrt{3})) \)**

- Now we need to find \( f(f(\sqrt{3})) \). From Step 1, we found that \( f(\sqrt{3}) = 0 \).
- Next, we evaluate \( f(0) \).

We need to check if \( 0 \) is a rational or an irrational number. Since \( 0 \) can be expressed as \( \frac{0}{1} \), it is a **rational number**.

Thus, according to the function definition:
\[ f(0) = 1 \quad \text{(since } 0 \text{ is rational)} \]

**Final Result:**
\[ f(f(\sqrt{3})) = f(0) = 1 \]

### Conclusion:
The value of \( f(f(\sqrt{3})) \) is \( 1 \).

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