Questions: Complete the description of the piecewise function graphed below. I if -6 ≤ x ≤ -2 f(x)=1 if -2 < x ≤ 1 1 if 1 < x ≤ 6

Solution Steps

The line segment connecting (-6, 1) and (-2, 4) has a slope of (4-1)/(-2 - -6) = 3/4. Using the point-slope form of a linear equation, we have y - 1 = (3/4)(x - -6), simplifying to y = (3/4)x + 11/2.

The line segment is a horizontal line at y=4, hence the equation is y = 4. Note that x=-2 is excluded from this interval based on the open circle at x=-2.

The line segment connects (1, 4) and (6,-4). Its slope is (-4-4)/(6-1) = -8/5. Using point-slope form, we have y - 4 = (-8/5)(x-1). Simplifying, we get y = (-8/5)x + 28/5. Note that x=1 is excluded from this interval due to the open circle at x=1.

Final Answer:

f(x) = { (3/4)x + 11/2   if -6 ≤ x ≤ -2
         { 4                if -2 < x ≤ 1
         { (-8/5)x + 28/5  if 1 < x ≤ 6