Mailing a first-class letter costs 0.46 for the first ounce and 0.20 for each additional ounce or portion of an ounce. If y is the total cost of mailing a letter weighing x ounces, which function model fits: step, square root, linear, or absolute value?

Step-by-step solution

1. 1

   Find the phrase that controls the answer. It is or portion of an ounce. A letter weighing $1.1$ oz is charged exactly the same as one weighing $2.0$ oz, because any part of the second ounce is billed as a whole ounce. Cost therefore depends on $\lceil x\rceil$, not on $x$ itself.

2. 2

   Tabulate the cost over the first few intervals.

   $0<x\le 1:\ y=0.46$

   $1<x\le 2:\ y=0.46+0.20=0.66$

   $2<x\le 3:\ y=0.46+0.40=0.86$

   Within each interval the value never changes; at each whole-ounce boundary it jumps by $0.20$.

3. 3

   Match that behaviour to a function type. A graph made of horizontal segments separated by sudden vertical jumps is the definition of a step function. In closed form:

   $y=0.46+0.20\left(\lceil x\rceil-1\right),\qquad x>0$

4. 4

   Rule out the other three options. A linear model $y=0.46+0.20(x-1)$ would charge $\\$0.48$for a$1.1$oz letter, but the post office charges$\$0.66$ — linear growth is continuous, and this cost is not. A square root model grows without bound but with a decreasing rate, which nothing here suggests. An absolute value model is V-shaped, meaning cost would fall and then rise; postage never falls as weight increases.

5. 5

   State the conclusion. The appropriate model is a step function, the standard model for any price that is charged per started unit — parking by the hour, data by the gigabyte, or postage by the ounce.