The function you've provided is:

\[
f(x) = |x - 3| + 1
\]

This is a piecewise function because it includes the absolute value term, which behaves differently depending on whether \( x - 3 \) is positive or negative.

To analyze the function in detail:

### 1. **When \( x \geq 3 \):**
If \( x \geq 3 \), then \( |x - 3| = x - 3 \). The function becomes:
\[
f(x) = (x - 3) + 1 = x - 2
\]

### 2. **When \( x < 3 \):**
If \( x < 3 \), then \( |x - 3| = -(x - 3) = 3 - x \). The function becomes:
\[
f(x) = (3 - x) + 1 = 4 - x
\]

### Piecewise Function:
Thus, the function can be written as:
\[
f(x) =
\begin{cases}
x - 2 & \text{if } x \geq 3 \\
4 - x & \text{if } x < 3
\end{cases}
\]

### Key Points:
- **Vertex**: The "kink" of the absolute value function happens at \( x = 3 \), where the two pieces of the function meet. At \( x = 3 \), \( f(3) = |3 - 3| + 1 = 1 \), so the point \( (3, 1) \) is the vertex of the function.
- **Slope**: For \( x \geq 3 \), the slope is 1. For \( x < 3 \), the slope is -1.
- **Graph**: The graph will have a V-shape with its vertex at \( (3, 1) \). The left side decreases with a slope of -1, and the right side increases with a slope of 1.

Would you like a graphical representation or more details?

### 5 Related Questions:
1. How do you graph a piecewise function?
2. What is the significance of the absolute value function in real-world contexts?
3. How do you find the domain and range of this function?
4. What are the transformations applied to the basic \( |x| \) function in \( f(x) = |x - 3| + 1 \)?
5. How do you solve equations involving absolute value functions?

### Tip:
When graphing absolute value functions, always locate the vertex first, as it defines the point where the function changes its direction.

Explore the detailed solution to the absolute value function f(x) = |x − 3| + 1. This step-by-step guide covers key algebraic concepts such as piecewise functions, absolute value properties, and the graphical representation of the function. Perfect for students in Grades 8-10.

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