graph the function.\nf(x) = {3x + 2, x < -2; x, -2 ≤ x ≤ 3; 2x - 1, x > 3}

Explanation:

Step1: Analyze (y = 3x+2) for (x < - 2)

Find two - point form. When (x=-3), (y=3\times(-3)+2=-9 + 2=-7). The line (y = 3x + 2) has a slope of (3) and a (y) - intercept of (2), but we only consider the part where (x < - 2). The point ((-3,-7)) is on this part of the line, and we draw an open - circle at (x=-2), (y=3\times(-2)+2=-4).

Step2: Analyze (y = x) for (-2\leq x\leq3)

When (x=-2), (y=-2); when (x = 3), (y = 3). We draw a solid line segment connecting the points ((-2,-2)) and ((3,3)) since the inequality includes the endpoints.

Step3: Analyze (y=2x - 1) for (x>3)

When (x = 4), (y=2\times4-1=7). The line (y = 2x-1) has a slope of (2) and a (y) - intercept of (-1), and we draw an open - circle at (x = 3), (y=2\times3-1=5).

To graph the function, we combine these three parts on the same coordinate plane.

Answer:

The graph consists of a part of the line (y = 3x+2) for (x < - 2) (open - circle at (x=-2)), a line segment (y = x) for (-2\leq x\leq3) (solid endpoints), and a part of the line (y=2x - 1) for (x>3) (open - circle at (x = 3)).