To solve this problem, we need to understand the behavior of the function \(f(x) = ||x + 2| - |x - 1||\) and match it with one of the given graphs. Let's analyze the function step by step:

Step 1: Break Down the Absolute Value Expressions

• The function involves absolute values, which means it will have different expressions based on the sign of the expressions inside the absolute values.
• Consider the critical points where the expressions inside the absolute values change sign: \(x = -2\) and \(x = 1\).

Step 2: Evaluate the Function in Different Intervals

• \(x \leq -2\): Both \(x + 2\) and \(x - 1\) are negative.
• The function simplifies to: \(f(x) = |-(x + 2) - (-(x - 1))| = |-x - 2 + x - 1| = |(-3)| = 3\).
• \(-2 \lt x \leq 1\): \(x + 2\) is positive, \(x - 1\) is negative.
• The function simplifies to: \(f(x) = |(x + 2) - (-(x - 1))| = |x + 2 + x - 1| = |2x + 1|\).
• \(x \gt 1\): Both \(x + 2\) and \(x - 1\) are positive.
• The function simplifies to: \(f(x) = |(x + 2) - (x - 1)| = |x + 2 - x + 1| = |3| = 3\).

Step 3: Sketch the Graph

• In the intervals \(x \leq -2\) and \(x \gt 1\), the function is constant with a value of 3.
• In the interval \(-2 \lt x \leq 1\), the function is a linear function \(|2x + 1|\), which forms a 'V' shape at \(x = -\frac{1}{2}\).

Conclusion

• Based on the analysis above, graph "P" displays constant values of 3 for \(x \leq -2\) and \(x \gt 1\), and a 'V' shape between \(x = -2\) and \(x = 1\).

Thus, the correct graph that represents the function is graph "P".