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 \( -1 \leq x \leq 3 \), we can analyze the function by breaking it into different cases based on the values of \( x \) where the expressions inside the absolute values change sign.

### Step 1: Identify critical points
The critical points where the expression inside the absolute values changes are \( x = 1 \) and \( x = 2 \). We will analyze the function in three intervals:
1. \( -1 \leq x < 1 \)
2. \( 1 \leq x < 2 \)
3. \( 2 \leq x \leq 3 \)

### Step 2: Analyze the first interval \( -1 \leq x < 1 \)
In this interval, both \( x - 1 \) and \( x - 2 \) are negative. Therefore, we can rewrite the function as:
\[ f(x) = -(x - 1) - (x - 2) = -x + 1 - x + 2 = -2x + 3 \]
Now, we will evaluate \( f(x) \) at the endpoints of this interval:
- At \( x = -1 \):
\[ f(-1) = -2(-1) + 3 = 2 + 3 = 5 \]
- As \( x \) approaches \( 1 \) from the left:
\[ f(1) = -2(1) + 3 = -2 + 3 = 1 \]

### Step 3: Analyze the second interval \( 1 \leq x < 2 \)
In this interval, \( x - 1 \) is non-negative and \( x - 2 \) is negative. Thus, we can rewrite the function as:
\[ f(x) = (x - 1) - (x - 2) = x - 1 - x + 2 = 1 \]
This means that \( f(x) \) is constant and equal to \( 1 \) for all \( x \) in this interval.

### Step 4: Analyze the third interval \( 2 \leq x \leq 3 \)
In this interval, both \( x - 1 \) and \( x - 2 \) are non-negative. Therefore, we can rewrite the function as:
\[ f(x) = (x - 1) + (x - 2) = x - 1 + x - 2 = 2x - 3 \]
Now, we will evaluate \( f(x) \) at the endpoints of this interval:
- At \( x = 2 \):
\[ f(2) = 2(2) - 3 = 4 - 3 = 1 \]
- At \( x = 3 \):
\[ f(3) = 2(3) - 3 = 6 - 3 = 3 \]

### Step 5: Combine results
Now we can summarize the values of \( f(x) \) in the three intervals:
- For \( -1 \leq x < 1 \), \( f(x) \) ranges from \( 5 \) to \( 1 \).
- For \( 1 \leq x < 2 \), \( f(x) = 1 \).
- For \( 2 \leq x \leq 3 \), \( f(x) \) ranges from \( 1 \) to \( 3 \).

### Step 6: Determine the overall range
The minimum value of \( f(x) \) is \( 1 \) and the maximum value is \( 5 \). Therefore, the range of the function \( f(x) \) is:
\[ \text{Range of } f(x) = [1, 5] \]