Question

1. write a piecewise function for the graph below. (4pts)

Explanation:

Step1: Analyze the lower segment (x ≤ 1)

The lower line passes through (0, -4)? Wait, no, looking at the grid, the lower line has a point at (1, -3)? Wait, no, the solid dot is at (1, -3)? Wait, the y-intercept: when x=0, the line passes through (0, -4)? Wait, no, let's check the slope. From the left, the line goes through (0, -4) and (1, -3)? Wait, no, the solid dot is at (1, -3)? Wait, maybe I misread. Wait, the lower segment: let's find two points. Let's see, when x=0, y=-4? No, the line passes through (0, -4) and (1, -3)? Wait, slope is ( -3 - (-4) ) / (1 - 0) = 1/1 = 1? Wait, no, the solid dot is at (1, -3), and the line goes to the left. Wait, maybe the lower segment is for x ≤ 1, with slope 1, and y-intercept -4? Wait, when x=0, y=-4? Then the equation is y = x - 4? Wait, no, when x=1, y = 1 - 4 = -3, which matches the solid dot. So for x ≤ 1, the function is y = x - 4? Wait, no, the solid dot is at (1, -3), so when x=1, y=-3. So the equation for the lower line: slope m = (y2 - y1)/(x2 - x1). Let's take (0, -4) and (1, -3): m = (-3 - (-4))/(1 - 0) = 1/1 = 1. So equation is y = x - 4, with domain x ≤ 1 (since the solid dot is at x=1, so included).

Step2: Analyze the upper segment (x > 1)

The upper segment has an open circle at (1, 3) (wait, the open circle is at (1, 3)? Wait, the grid: x=1, y=3? Wait, the open circle is at (1, 3), and the line goes up with slope. Let's find two points. The open circle is at (1, 3), and another point: (2, 6)? Wait, from (1, 3) to (2, 6), slope is (6 - 3)/(2 - 1) = 3/1 = 3. So the equation is y - 3 = 3(x - 1), which simplifies to y = 3x - 3 + 3 = 3x? Wait, no, 3(x - 1) + 3 = 3x - 3 + 3 = 3x? Wait, when x=1, y=3(1)=3, which matches the open circle. Then when x=2, y=6, which is on the line. So for x > 1, the function is y = 3x, with domain x > 1 (open circle at x=1, so not included).

Step3: Combine into piecewise function

So the piecewise function is:

$$ f(x) =
LATEXBLOCK0
$$

Wait, but let's check the open circle: at x=1, the upper function has open circle at (1, 3), and the lower function has solid circle at (1, -3). Wait, maybe I messed up the lower line. Wait, the solid dot is at (1, -3), and the line goes to the left. Let's re-examine the graph. The lower line: starting from the left, going up, passing through (0, -4) and (1, -3), then the upper line starts at open circle (1, 3) and goes up with slope 3. So yes, the lower segment is for x ≤ 1, y = x - 4 (since at x=1, y=1 - 4 = -3, which is the solid dot), and the upper segment is for x > 1, y = 3x (at x=1, y=3(1)=3, which is the open circle).

Answer:

$$ f(x) =
LATEXBLOCK0
$$