`f(x)=abs(x-1)+abs(x-2)+abs(x-3)` . Find the range of f(x).

Answer

To find the range of the function \( f(x) = |x-1| + |x-2| + |x-3| \), we will analyze it by breaking it down into different cases based on the value of \( x \).

### Step 1: Identify critical points
The critical points for the function occur at \( x = 1 \), \( x = 2 \), and \( x = 3 \). We will evaluate the function in the intervals defined by these points:
1. \( x < 1 \)
2. \( 1 \leq x < 2 \)
3. \( 2 \leq x < 3 \)
4. \( x \geq 3 \)

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

**Case 1: \( x < 1 \)**
In this case, all expressions inside the absolute values are negative:
\[ f(x) = -(x-1) - (x-2) - (x-3) = -x + 1 - x + 2 - x + 3 = -3x + 6 \]
This is a linear function with a slope of -3.

**Case 2: \( 1 \leq x < 2 \)**
Here, \( |x-1| \) is positive and the others are negative:
\[ f(x) = (x-1) - (x-2) - (x-3) = x - 1 - x + 2 - x + 3 = -x + 4 \]
This is also a linear function, but with a slope of -1.

**Case 3: \( 2 \leq x < 3 \)**
In this interval, \( |x-1| \) and \( |x-2| \) are positive, while \( |x-3| \) is negative:
\[ f(x) = (x-1) + (x-2) - (x-3) = x - 1 + x - 2 - x + 3 = x \]
This is a linear function with a slope of 1.

**Case 4: \( x \geq 3 \)**
In this case, all expressions are positive:
\[ f(x) = (x-1) + (x-2) + (x-3) = x - 1 + x - 2 + x - 3 = 3x - 6 \]
This is a linear function with a slope of 3.

### Step 3: Determine the values at critical points
Now we will evaluate \( f(x) \) at the critical points to find the minimum value:
- For \( x = 1 \):
\[ f(1) = |1-1| + |1-2| + |1-3| = 0 + 1 + 2 = 3 \]
- For \( x = 2 \):
\[ f(2) = |2-1| + |2-2| + |2-3| = 1 + 0 + 1 = 2 \]
- For \( x = 3 \):
\[ f(3) = |3-1| + |3-2| + |3-3| = 2 + 1 + 0 = 3 \]

### Step 4: Analyze the function behavior
- As \( x \to -\infty \), \( f(x) \to \infty \) (from Case 1).
- At \( x = 2 \), \( f(2) = 2 \) is the minimum value.
- As \( x \to \infty \), \( f(x) \to \infty \) (from Case 4).

### Conclusion
The minimum value of \( f(x) \) is 2, and since the function tends to infinity in both directions, the range of \( f(x) \) is:
\[ \text{Range of } f(x) = [2, \infty) \]