Question

which graph is the graph of this function?
$f(x) = \

$$\begin{cases} 5 & \\text{if } -5 \\leq x < -2 \\\\ -1 & \\text{if } -2 \\leq x < 3 \\\\ 1 & \\text{if } 3 \\leq x \\leq 4 \\end{cases}$$

$
a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Analyze the piecewise function

The function \( f(x) \) has three pieces:

• For \( -3 \leq x < -1 \), \( f(x) = 5 \) (a horizontal line at \( y = 5 \)).
• For \( -2 \leq x < 3 \), \( f(x) = -1 \) (a horizontal line at \( y = -1 \)) – wait, correction: looking at the options, maybe the intervals are \( -3 \leq x < -1 \) (y=5), \( -2 \leq x < 3 \) (y=-1)? Wait, no, re - check the function (from the image, the function is \( f(x)=

$$\begin{cases}5&\text{if } - 3\leq x < - 1\\-1&\text{if } - 2\leq x < 3\\1&\text{if } 3\leq x\leq4\end{cases}$$

\) (assuming the last interval is \( 3\leq x\leq4 \) with \( y = 1 \)).

Step2: Check each graph

• Graph A: Check the intervals. For the first piece (\( y = 5 \)): interval \( -3\leq x < -1 \)? No, the first horizontal line in graph A doesn't match.
• Graph B:
• First piece: \( y = 5 \) for \( -3\leq x < -1 \): The horizontal line at \( y = 5 \) is between \( x=-3 \) and \( x = - 1 \) (matches).
• Second piece: \( y=-1 \) for \( -2\leq x < 3 \): The horizontal line at \( y=-1 \) is between \( x = - 2 \) and \( x = 3 \) (matches).
• Third piece: \( y = 1 \) for \( 3\leq x\leq4 \): The horizontal line at \( y = 1 \) is between \( x = 3 \) and \( x = 4 \) (matches).
• Graph C: The intervals for the horizontal lines do not match the function's intervals.
• Graph D: The intervals and the \( y \) - values for the horizontal lines do not match the function's definition.

Answer:

B. graph B