The range of the function `f(x)=|x-1|+|x-2|, -1 le x le 3,` is

Options

A. [1, 3]
B. [1, 5]
C. [3, 5]
D. None of these

Explanation

To find the range of the function \( f(x) = |x - 1| + |x - 2| \) for the interval \( -1 \leq x \leq 3 \), we will analyze the function by breaking it down based on the critical points where the expression inside the absolute values changes sign. The critical points here are \( x = 1 \) and \( x = 2 \).

### Step 1: Identify the intervals
The critical points divide the interval \( [-1, 3] \) into three segments:
1. \( x \in [-1, 1] \)
2. \( x \in [1, 2] \)
3. \( x \in [2, 3] \)

### Step 2: Evaluate \( f(x) \) in each interval

**Interval 1: \( x \in [-1, 1] \)**
- In this interval, both \( x - 1 \) and \( x - 2 \) are negative.
- Therefore, we have:
\[ f(x) = -(x - 1) - (x - 2) = -x + 1 - x + 2 = -2x + 3 \]
- Evaluating at the endpoints:
- At \( x = -1 \):
\[ f(-1) = -2(-1) + 3 = 2 + 3 = 5 \]
- At \( x = 1 \):
\[ f(1) = -2(1) + 3 = -2 + 3 = 1 \]

**Interval 2: \( x \in [1, 2] \)**
- In this interval, \( x - 1 \) is non-negative and \( x - 2 \) is negative.
- Therefore, we have:
\[ f(x) = (x - 1) - (x - 2) = x - 1 - x + 2 = 1 \]
- Here, \( f(x) \) is constant and equal to 1 for all \( x \) in this interval.

**Interval 3: \( x \in [2, 3] \)**
- In this interval, both \( x - 1 \) and \( x - 2 \) are non-negative.
- Therefore, we have:
\[ f(x) = (x - 1) + (x - 2) = x - 1 + x - 2 = 2x - 3 \]
- Evaluating at the endpoints:
- At \( x = 2 \):
\[ f(2) = 2(2) - 3 = 4 - 3 = 1 \]
- At \( x = 3 \):
\[ f(3) = 2(3) - 3 = 6 - 3 = 3 \]

### Step 3: Compile the results
From our evaluations:
- In the interval \( [-1, 1] \), \( f(x) \) varies from \( 5 \) to \( 1 \).
- In the interval \( [1, 2] \), \( f(x) = 1 \).
- In the interval \( [2, 3] \), \( f(x) \) varies from \( 1 \) to \( 3 \).

### Step 4: Determine the range
Combining all the values, we see that:
- The minimum value of \( f(x) \) is \( 1 \) (which occurs in both the intervals \( [1, 2] \) and \( [2, 3] \)).
- The maximum value of \( f(x) \) is \( 5 \) (which occurs at \( x = -1 \)).

Thus, the range of the function \( f(x) \) is:
\[ \text{Range} = [1, 5] \]