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:

\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.