Complete the equation for the piecewise function graphed bel

Complete the equation for the piecewise function graphed below. f(x) = { if -6

Solution

!. If - 6 x -3. The piecewise function in this interval is a straight line, which passes through the points ( - 6, 3) and ( -3, - 3) . Let the equation of the required line be y = mx + c. On substituting x = - 6 and y = 3, we get 3 = - 6m + c. ..(1) . Also, on substituting x = - 3 and y = - 3, we get -3 = -m + c...(2) On subtracting the 2nd equation from the 1st equation, we get 3 - ( -3) = - 6m + c - (-3m) - c or, 6 = -3m so that m = - 2. On substituting m = -2 in the 1st equation, we get 3 = -6 (-2) + c or, c = 3 - 12 = -9. Thus, the equation of the line is y = - 2x - 9.

2. If -3 < x 1. In this interval, the piecewise function is a straight line which passes through the points ( 0,-4) and (1, -4). Let the equation of the line be y = mx + c. On substituting x = 0 and y = - 4, we get -4 = c. ..(1) . Also, on substituting x = 1 and y = - 4, we get - 4 = m + c...(2) Since c = -4, we have from the 2nd equation, m - 4 = -4 so that m = 0. Then the equation of the line is y = -4.

3. If 1 < x 6. In this interval, the piecewise function is a straight line which passes through the points ( 3,- 4) and (5, - 3). Let the equation of the line be y = mx + c. On substituting x = 3 and y = - 4, we get -4 = 3m + c. ..(1) . Also, on substituting x = 5 and y = - 3, we get - 3 = 5m + c...(2) Now on subtractiong the 1st equation from the 2nd equation, we get 5m + c - 3m - c = -3 - (-4) or, 2m = 1 so that m = 1/2. Now, on substituting m = 1/2 in the 1st equation, we get -4 = 3(1/2) + c so that c = -4 - 3/2 = -11/2. Then the equation of the line is y = x/2 - 11/2.

The piecewise function is y = f(x ) = - 2x - 9. if - 6 x -3.

y = f ( x) = - 4 if 1 < x 6

y = f (x ) = x/2 - 11/2. if 1 < x 6.