Find the range of each of the following functions. (i) `f(x) = 2 3x , x in R , 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) Function: \( f(x) = 2 \cdot 3^x \), where \( x > 0 \)

1. **Understanding the function**: The function \( f(x) = 2 \cdot 3^x \) is an exponential function. The base \( 3 \) is greater than \( 1 \), which means the function will increase as \( x \) increases.

2. **Evaluating the function at the lower limit**: Since \( x > 0 \), we can evaluate \( f(x) \) at \( x = 0 \):
\[ f(0) = 2 \cdot 3^0 = 2 \cdot 1 = 2 \]

3. **Behavior as \( x \) approaches infinity**: As \( x \) increases towards infinity, \( 3^x \) also increases towards infinity:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} 2 \cdot 3^x = \infty \]

4. **Determining the range**: Since \( f(x) \) starts from \( 2 \) (when \( x \) is just greater than \( 0 \)) and goes to infinity, the range of \( f(x) \) is:
\[ \text{Range} = (2, \infty) \]