Question

Which one of the given figures P, Q, R, and S represents the graph of the following function? \[ f(x) = |x + 2| - |x - 1| \]

• A. P
• B. Q
• C. R
• D. S

Hint:

- Absolute value functions break into piecewise linear sections, which can cause slope changes at certain points (where the inside expression equals zero). - To graph such functions, first identify these critical points and then plot accordingly in each region.

Answer 1

The Correct Option is A

We are given the function: \[ f(x) = |x + 2| - |x - 1| \] To determine the graph of this function, we need to analyze the behavior of each part of the function.
1) Function Analysis:
We can break the function into two absolute value terms. The behavior of the function will change depending on whether \( x \) is less than -2, between -2 and 1, or greater than 1. We need to analyze these intervals to plot the graph.
- For \( x<-2 \), both \( x + 2 \) and \( x - 1 \) are negative, so the graph will have a linear decrease.
- For \( -2 \leq x \leq 1 \), \( x + 2 \) is positive, and \( x - 1 \) is negative, resulting in a change in slope and a turning point.
- For \( x>1 \), both terms are positive, resulting in a different slope.
2) Matching the Graph:
Upon examining the graphs, the graph labeled \( P \) fits the expected behavior of the function with changes in slope at \( x = -2 \) and \( x = 1 \). Thus, the correct graph is option (A).
\[ \boxed{\text{The correct answer is (A) P.}} \]