A function f is defined on the set of real numbers as follows:
`f(x)={(x+1, 1 le x lt 2),(2x-1, 2 le x lt 4), (3x-10, 4 le x lt 6):}`
(a) Find the domain of the function.
(b) Find the range of the function.

Answer

To solve the problem, we will break it down into two parts: finding the domain and finding the range of the function \( f(x) \).

### Part (a): Finding the Domain of the Function

1. **Identify the intervals of the function**: The function \( f(x) \) is defined in three intervals:
- \( f(x) = x + 1 \) for \( 1 \leq x < 2 \)
- \( f(x) = 2x - 1 \) for \( 2 \leq x < 4 \)
- \( f(x) = 3x - 10 \) for \( 4 \leq x < 6 \)

2. **Determine the endpoints**: The function is defined at the left endpoints of each interval and does not include the right endpoints. The critical points where the function changes its definition are \( x = 1, 2, 4, \) and \( 6 \).

3. **Combine the intervals**: The function is defined for all \( x \) in the intervals:
- From \( 1 \) to \( 2 \) (including \( 1 \) but not \( 2 \))
- From \( 2 \) to \( 4 \) (including \( 2 \) but not \( 4 \))
- From \( 4 \) to \( 6 \) (including \( 4 \) but not \( 6 \))

4. **Write the domain**: Therefore, the domain of the function is:
\[ \text{Domain} = [1, 6) \]

### Part (b): Finding the Range of the Function

1. **Evaluate the function at the endpoints of each interval**:
- For \( x = 1 \):
\[ f(1) = 1 + 1 = 2 \]
- For \( x = 2 \):
\[ f(2) = 2(2) - 1 = 3 \]
- For \( x = 4 \):
\[ f(4) = 3(4) - 10 = 2 \]
- For \( x = 6 \):
\[ f(6) = 3(6) - 10 = 8 \quad (\text{but } x = 6 \text{ is not included in the domain}) \]

2. **Determine the range for each interval**:
- For \( 1 \leq x < 2 \):
- As \( x \) approaches \( 2 \), \( f(x) \) approaches \( 3 \). Thus, the range is \( [2, 3) \).
- For \( 2 \leq x < 4 \):
- As \( x \) approaches \( 4 \), \( f(x) \) approaches \( 7 \). Thus, the range is \( [3, 7) \).
- For \( 4 \leq x < 6 \):
- As \( x \) approaches \( 6 \), \( f(x) \) approaches \( 8 \). Thus, the range is \( [2, 8) \).

3. **Combine the ranges**: The overall range of the function combines all these intervals:
\[ \text{Range} = [2, 3) \cup [3, 7) \cup [2, 8) = [2, 8) \]

### Final Answers

- **Domain**: \( [1, 6) \)
- **Range**: \( [2, 8) \)