Question

which piecewise function best describes the scenario below?

a shipping company charges for shipping based on weight of package:

• for packages weighing up to 5 pounds, the cost is $4.
• for packages weighing more than 5 but less than 10 pounds, the cost is $8.
• for packages weighing 10 pounds or more, the cost is $12.

option 1:
\\( f(x) = \

$$\begin{cases} 4 & 0 \\le x \\le 5 \\\\ 8 & 5 < x < 10 \\\\ 12 & x \\ge 10 \\end{cases}$$

\\)

option 2:
\\( f(x) = \

$$\begin{cases} 4 & 0 \\le x \\le 5 \\\\ 8 & 5 \\le x \\le 10 \\\\ 12 & x \\ge 10 \\end{cases}$$

\\)

option 3:
\\( f(x) = \

$$\begin{cases} 4 & 0 < x < 5 \\\\ 8 & 5 < x < 10 \\\\ 12 & x > 10 \\end{cases}$$

\\)

option 4:
\\( f(x) = \

$$\begin{cases} 12 & 0 \\le x \\le 5 \\\\ 8 & 5 < x < 10 \\\\ 4 & x \\ge 10 \\end{cases}$$

\\)

Explanation:

Translate the first condition into an inequality

Using the Piecewise Functions knowledge point

• "Up to 5 pounds" means the weight \(x\) is greater than or equal to 0 and less than or equal to 5.
• This corresponds to the interval \(0 \le x \le 5\) with a cost of 4.

Translate the second condition into an inequality

Using the Piecewise Functions knowledge point

• "More than 5 but less than 10 pounds" means the weight \(x\) is strictly greater than 5 and strictly less than 10.
• This corresponds to the interval \(5 < x < 10\) with a cost of 8.

Translate the third condition into an inequality

Using the Piecewise Functions knowledge point

• "10 pounds or more" means the weight \(x\) is greater than or equal to 10.
• This corresponds to the interval \(x \ge 10\) with a cost of 12.

Match the intervals to the options

Using the Piecewise Functions knowledge point

• Combining these three intervals gives:

$$ F(x) =
LATEXBLOCK0
$$

• This matches the first option.

Answer:

• **(A) \(F(x) =

$$\begin{cases} 4 & 0 \le x \le 5 \\ 8 & 5 < x < 10 \\ 12 & x \ge 10 \end{cases}$$

\) (Correct answer)**

• (B) \(F(x) =

$$\begin{cases} 4 & 0 \le x \le 5 \\ 8 & 5 \le x \le 10 \\ 12 & x \ge 10 \end{cases}$$

\)

• (C) \(F(x) =

$$\begin{cases} 4 & 0 < x < 5 \\ 8 & 5 < x < 10 \\ 12 & x > 10 \end{cases}$$

\)

• (D) \(F(x) =

$$\begin{cases} 12 & 0 \le x \le 5 \\ 8 & 5 < x < 10 \\ 4 & x \ge 10 \end{cases}$$

\)