Range of f(x) = |x-3| is

Options

A. `(-oo, oo)`
B. `(-oo, 3]`
C. `(-3, 3)`
D. `[0, oo)`

Answer

To find the range of the function \( f(x) = |x - 3| \), we can follow these steps:

### Step 1: Understand the Absolute Value Function
The function \( f(x) = |x - 3| \) represents the absolute value of \( x - 3 \). The absolute value function always yields non-negative results.

### Step 2: Determine the Behavior of the Function
The function can be broken down into two cases based on the value of \( x \):
- When \( x \geq 3 \):
\[ f(x) = x - 3 \]
- When \( x < 3 \):
\[ f(x) = -(x - 3) = 3 - x \]

### Step 3: Analyze Each Case
1. **For \( x \geq 3 \)**:
- As \( x \) increases from 3 to infinity, \( f(x) = x - 3 \) will also increase from 0 to infinity.
- Therefore, \( f(x) \) takes values from 0 to \( \infty \).

2. **For \( x < 3 \)**:
- As \( x \) decreases from 3 to negative infinity, \( f(x) = 3 - x \) will also increase.
- When \( x \) approaches 3 from the left, \( f(x) \) approaches 0.
- As \( x \) decreases further, \( f(x) \) will take on all positive values.

### Step 4: Combine the Results
From both cases, we see that:
- The minimum value of \( f(x) \) is 0 (when \( x = 3 \)).
- There is no upper limit to the values \( f(x) \) can take as \( x \) moves away from 3.

### Conclusion
Thus, the range of the function \( f(x) = |x - 3| \) is:
\[ [0, \infty) \]