Find the range of each of the following functions.
(i) `f(x) = 2 - 3x , x in mathbbR , x > 0`
(ii) `f(x) =x^2+ 2`, x is a real number.
(iii) `f(x) = x`, `x` is a real number.

Explanation

To find the range of the given functions, we will analyze each function step by step.

### (i) \( f(x) = 2 - 3x \), where \( x \in \mathbb{R} \) and \( x > 0 \)

1. **Identify the function**: The function is linear and can be expressed as \( f(x) = 2 - 3x \).
2. **Determine the domain**: The domain is \( x > 0 \).
3. **Evaluate the function at the boundary of the domain**:
- As \( x \) approaches 0 from the right, \( f(x) \) approaches \( 2 - 3(0) = 2 \).
- As \( x \) increases towards infinity, \( f(x) \) approaches \( 2 - 3(\infty) = -\infty \).
4. **Determine the range**: Since \( f(x) \) decreases from \( 2 \) to \( -\infty \) as \( x \) increases from \( 0 \) to \( \infty \), the range is \( (-\infty, 2) \).