$$g(x) = \\begin{cases} -3x - 8 &, -4 \\leq x \\leq 0 \\\\ -3x + 9 &, 0 < x \\leq 3 \\end{cases}$$ what is…

Explanation:

Step1: Analyze the first piece of the function

For ( g(x) = -3x - 8 ) with ( -4 \leq x \leq 0 ):

• When ( x = -4 ), ( g(-4)=-3(-4)-8 = 12 - 8 = 4 ). So the point is ( (-4, 4) ).
• When ( x = 0 ), ( g(0)=-3(0)-8=-8 ). So the point is ( (0, -8) ).

Step2: Analyze the second piece of the function

For ( g(x) = -3x + 9 ) with ( 0 < x \leq 3 ):

• When ( x = 0 ) (approaching from the right), ( g(0^+)=-3(0)+9 = 9 ). But at ( x = 0 ), the first piece gives ( -8 ), so there's a jump at ( x = 0 ).
• When ( x = 3 ), ( g(3)=-3(3)+9 = 0 ). So the point is ( (3, 0) ).

Now, check the graphs:

• Graph A: At ( x = 2 ) (in ( 0 < x \leq 3 )), the y - value should be ( -3(2)+9 = 3 ), but the graph in A at ( x = 2 ) has a different y - value.
• Graph B: For the first piece, when ( x=-4 ), ( y = 4 ) (matches), at ( x = 0 ), ( y=-8 ) (matches). For the second piece, at ( x = 3 ), ( y = 0 ) (matches), and the slope of both pieces is - 3 (since the coefficient of x is - 3 in both linear functions), so the lines are parallel (same slope) with a jump at ( x = 0 ) from ( y=-8 ) (first piece) to ( y = 9 ) (second piece at ( x = 0^+ )).

Answer:

B