Answer: { (-1/2)x - 1, -2 ≤ x ≤ 4
Explanation:
Let's start with the first line:
The line passes through two points: (-5, 8) and (-2, 2). The hollow circle at (-5, 8) indicates that this point is not included in the graph, while the closed circle at (-2, 2) indicates that this point is included.
We can find the slope of the line using the two points:
slope = (change in y) / (change in x) = (2 - 8) / (-2 - (-5)) = -6 / 3 = -2
Using point-slope form, we can write the equation of the line as:
y - 2 = -2(x - (-2))
Simplifying:
y - 2 = -2x - 4
y = -2x - 2
Next, let's consider the second line:
The line passes through two points: (-2, -2) and (4, -5). The hollow circle at (-2, -2) indicates that this point is not included in the graph, while the closed circle at (4, -5) indicates that this point is included.
We can find the slope of the line using the two points:
slope = (change in y) / (change in x) = (-5 - (-2)) / (4 - (-2)) = -3 / 6 = -1/2
Using point-slope form, we can write the equation of the line as:
y - (-2) = (-1/2)(x - (-2))
Simplifying:
y + 2 = (-1/2)x + 1
y = (-1/2)x - 1
Now we can write the piecewise function:
f(x) = { -2x - 2, -5 ≤ x < -2
{ (-1/2)x - 1, -2 ≤ x ≤ 4
This piecewise function represents the two lines graphed on the given axes, where the first line is defined for x values between -5 and -2 (inclusive on -2 but not on -5), and the second line is defined for x values between -2 and 4 (inclusive on both).