Dayaan M. answered • 14d
Earned A’s Twice in Precalculus | 5 Years of Tutoring Experience
The whole problem turns on the word "max" in each column heading. The column headed 3 kg (max) means the package weighs more than 1 kg but not more than 3 kg, so every rate covers a whole band of weights rather than one exact weight. That is why the answer comes out as a step function instead of a formula you can plug into.
(a) The rule for NCR. The NCR row reads 120, 160, 220, 420, 820 across the bands capped at 1, 3, 5, 10 and 20 kg, so
C(x) = 120 if 0 < x ≤ 1
C(x) = 160 if 1 < x ≤ 3
C(x) = 220 if 3 < x ≤ 5
C(x) = 420 if 5 < x ≤ 10
C(x) = 820 if 10 < x ≤ 20
Look closely at how each band is written. It is open on the left and closed on the right, so a package weighing exactly 3 kg still pays 160 and not 220. If you flip those inequalities you end up charging a 3 kg package the 5 kg rate, and that is the single most common mistake on this type of problem. The domain stops at 20 because the table says 20 kg is the most the company will carry.
(b) The graph. Draw five horizontal segments at heights 120, 160, 220, 420 and 820. Since each band is open on the left and closed on the right, every segment gets an open circle at its left end and a filled circle at its right end. The first segment runs from just right of x = 0 over to a filled dot at (1, 120). The graph then jumps up to an open dot at (1, 160), runs across to a filled dot at (3, 160), and so on. The result is a staircase climbing to the right, and the steps get taller as you go, because the jumps are 40, then 60, then 200, then 400 pesos.
(c) C(12.3) for Mindanao. The Mindanao row is 150, 190, 370, 720, 1,420. Ask only one thing: which band does 12.3 land in? It is more than 10 and not more than 20, so it sits in the 20 kg band.
C(12.3) = 1,420 pesos
(d) C(2.2) for Luzon. The Luzon row is 150, 190, 320, 620, 1,220. Now 2.2 is more than 1 and not more than 3, so it sits in the 3 kg band.
C(2.2) = 190 pesos
A quick way to hold the whole idea in one sentence: you pay for the smallest weight limit in the table that your package does not go over. Writing it as bands the way part (a) does is just the mathematical version of that sentence, and once you see it that way both (c) and (d) become a matter of finding the right row and the right band instead of calculating anything.
