If `f (x)= {{:(x ^(3), , x =Q),(-x ^(3),,x ne Q):},` then :

Options

A. f (x) is periodic
B. f (x) is many-one
C. f (x) is one-one
D. range of the function is R

Answer

To solve the given problem, we need to analyze the function defined as:

\[ f(x) = \begin{cases} x^3 & \text{if } x \in \mathbb{Q} \\ -x^3 & \text{if } x \notin \mathbb{Q} \end{cases} \]

### Step-by-Step Solution:

1. **Understanding the Function**:
- The function \( f(x) \) behaves differently based on whether \( x \) is a rational number (denoted as \( \mathbb{Q} \)) or an irrational number (denoted as \( \mathbb{R} \setminus \mathbb{Q} \)).
- For rational \( x \), \( f(x) = x^3 \).
- For irrational \( x \), \( f(x) = -x^3 \).

2. **Graphing the Function**:
- To visualize the function, we can plot the graphs of \( y = x^3 \) and \( y = -x^3 \).
- The graph of \( y = x^3 \) is a cubic curve that passes through the origin and rises steeply in the first and third quadrants.
- The graph of \( y = -x^3 \) is also a cubic curve, but it falls steeply in the first quadrant and rises in the third quadrant.

3. **Analyzing the Options**:
- **Option 1: Is \( f(x) \) periodic?**
- A function is periodic if there exists a positive number \( T \) such that \( f(x + T) = f(x) \) for all \( x \).
- Since \( f(x) \) does not repeat its values at regular intervals, it is not periodic. Therefore, this option is incorrect.

- **Option 2: Is \( f(x) \) a one-one function?**
- A function is one-one (injective) if different inputs produce different outputs.
- For any rational \( x \), \( f(x) = x^3 \) gives a unique output, and for any irrational \( x \), \( f(x) = -x^3 \) also gives a unique output. However, since both rational and irrational numbers can map to the same output (e.g., \( f(1) = 1 \) and \( f(\sqrt[3]{-1}) = -1 \)), the function is not one-one. Therefore, this option is incorrect.

- **Option 3: Is \( f(x) \) a many-one function?**
- A function is many-one if multiple inputs can produce the same output.
- As noted, both rational and irrational inputs can yield the same output value, hence this function is many-one. Therefore, this option is correct.

- **Option 4: What is the range of \( f(x) \)?**
- The outputs of \( f(x) \) can be any real number since \( x^3 \) covers all real numbers for rational \( x \) and \(-x^3\) covers all real numbers for irrational \( x \).
- Thus, the range of \( f(x) \) is all real numbers, which is \( \mathbb{R} \). Therefore, this option is correct.

### Final Conclusion:
- The correct options are:
- Option 3: The function is many-one.
- Option 4: The range of the function is all real numbers.