The problem involves the absolute value functions and requires analyzing their difference. Here is the logical breakdown behind the question:

1. Understanding the individual functions:

   • $y = |x + 1|$ and $y = |x - 3|$ are V-shaped graphs.
   • $|x + 1|$ has its vertex at $x = -1$, while $|x - 3|$ has its vertex at $x = 3$.
   • The slopes of the lines change around these vertices, as absolute value functions "reflect" below the x-axis.

2. The given function $f(x) = |x + 1| - |x - 3|$:

   • This function is a combination (specifically, a subtraction) of two absolute value functions.
   • To analyze $f(x)$, you need to understand where each absolute value changes behavior (at their vertices, $x = -1$ and $x = 3$).
   • Between and beyond these points, you can simplify the function into different linear pieces.

3. Breaking the domain into intervals:
   Since absolute values change their expression at their critical points, you need to consider the following intervals:

   • $x < -1$
   • $-1 \leq x < 3$
   • $x \geq 3$

   Within each interval, you would remove the absolute value signs and analyze $f(x)$ as a linear expression.

4. Finding the graph:

   • For each interval, simplify $f(x)$ into a linear form (lines).
   • Sketch the resulting lines for each interval on the graph.
   • Ensure that at the critical points $x = -1$ and $x = 3$, the graph is continuous (connected without breaks).

5. Combining the results:

   • Draw the piecewise-linear graph formed by these intervals.
   • Check the slopes and intercepts for accuracy.

This approach systematically constructs the graph by analyzing the behavior of the absolute value expressions over different domains.

Do you want me to solve the function step-by-step and sketch its graph?

Here are 5 related questions for further practice:

1. What is the general form of an absolute value function, and how does it affect the graph?
2. How do you determine the vertex of an absolute value function?
3. What happens when two absolute value functions are added or subtracted?
4. How do you determine the intervals where absolute values change their behavior?
5. How can you check for continuity in piecewise functions?

Tip: Always identify where absolute values "switch" by setting their expressions equal to zero. These points are critical for analyzing behavior.