A function `f: R to R` is defined as:
`f(x) ={{:(1,x in Q),(-1,x notinQ):}`
Evaluate each of the following:
(i) `f(2),f(pi)" "(ii)` Range of `f`
`(iii) f^(-1)(-1)" "(iv) f^(-1)(1)`

Answer

To solve the problem step by step, let's evaluate each part of the question based on the function defined as:

\[ f(x) = \begin{cases} 1 & \text{if } x \in \mathbb{Q} \\ -1 & \text{if } x \notin \mathbb{Q} \end{cases} \]

### (i) Evaluate \( f(2) \) and \( f(\pi) \)

1. **Evaluate \( f(2) \)**:
- Since \( 2 \) is a rational number (it can be expressed as \( \frac{2}{1} \)), we use the first case of the function definition.
- Therefore, \( f(2) = 1 \).

2. **Evaluate \( f(\pi) \)**:
- The number \( \pi \) is an irrational number (it cannot be expressed as a fraction of two integers).
- Thus, we use the second case of the function definition.
- Therefore, \( f(\pi) = -1 \).

**Final answers for part (i)**:
- \( f(2) = 1 \)
- \( f(\pi) = -1 \)

### (ii) Find the Range of \( f \)

- The function \( f(x) \) can only take two values based on whether \( x \) is rational or irrational.
- If \( x \) is rational, \( f(x) = 1 \).
- If \( x \) is irrational, \( f(x) = -1 \).
- Therefore, the range of \( f \) is \( \{-1, 1\} \).

**Final answer for part (ii)**:
- Range of \( f = \{-1, 1\} \)

### (iii) Find \( f^{-1}(-1) \)

- We need to find all \( x \) such that \( f(x) = -1 \).
- From the function definition, \( f(x) = -1 \) when \( x \) is not a rational number (i.e., \( x \) is irrational).
- Therefore, \( f^{-1}(-1) \) consists of all irrational numbers.

**Final answer for part (iii)**:
- \( f^{-1}(-1) \) is the set of all irrational numbers.

### (iv) Find \( f^{-1}(1) \)

- We need to find all \( x \) such that \( f(x) = 1 \).
- From the function definition, \( f(x) = 1 \) when \( x \) is a rational number.
- Therefore, \( f^{-1}(1) \) consists of all rational numbers.

**Final answer for part (iv)**:
- \( f^{-1}(1) \) is the set of all rational numbers.

### Summary of Answers:
- (i) \( f(2) = 1 \), \( f(\pi) = -1 \)
- (ii) Range of \( f = \{-1, 1\} \)
- (iii) \( f^{-1}(-1) \) is the set of all irrational numbers.
- (iv) \( f^{-1}(1) \) is the set of all rational numbers.