Question

The 2014 domestic postage rate for first-class letters weighing 3.5oz or less is 49 cents for the first ounce (or less), plus 21 cents for each additional ounce (or part of an ounce). Express the postage P as a piecewise defined function of the weight x of a letter, with $0<x\le 3.5$and sketch a graph of the function.

Short answer

$P\left(x\right)=\left\{\begin{array}{l}0.49\text{when}0<x\le 1\\ 0.70\text{when1}<x\le 2\\ 0.91\text{when2}<x\le 3\\ 1.12\text{when3}<x\le 3.5\end{array}\right.$

Step-by-step solution

Step 1. Given Information.

Given that, for first-class letters weighing 3.5oz or less is 49 cents for the first ounce (or less), plus 21 cents for each additional ounce (or part of an ounce).

Step 2. To formulate the postage function.

We know that, 1 cent = $0.01

Now,

When$0<x\le 1$, the Postage function be

$\begin{array}{l}P\left(x\right)=49\text{cents}\\ \text{=\$0.49}\\ \Rightarrow P\left(x\right)=\text{\$0.49}\end{array}$

As Given, for the first ounce (or less), plus 21 cents for each additional ounce (or part of an ounce).

When$1<x\le 2$, the Postage function be

$\begin{array}{l}P\left(x\right)=\text{\$0.49+21cents}\\ \text{=\$0.49+0.21}\\ \text{=\$0.70}\\ \Rightarrow P\left(x\right)=\text{\$0.70}\end{array}$

When$2<x\le 3$, the Postage function be

$\begin{array}{l}P\left(x\right)=\text{\$0.70+21cents}\\ \text{=\$0.70+0.21}\\ \text{=\$0.91}\\ \Rightarrow P\left(x\right)=\text{\$0.91}\end{array}$

When$3<x\le 3.5$, the Postage function be

$\begin{array}{l}P\left(x\right)=\text{\$0.91+21cents}\\ \text{=\$0.91+0.21}\\ \text{=\$1.12}\\ \Rightarrow P\left(x\right)=\text{\$1.12}\end{array}$

Step 3. The formula for the Postage function P.

$P\left(x\right)=\left\{\begin{array}{l}0.49\text{when}0<x\le 1\\ 0.70\text{when1}<x\le 2\\ 0.91\text{when2}<x\le 3\\ 1.12\text{when3}<x\le 3.5\end{array}\right.$

Step 4. Graph of the Postage function P.