Question

given the piecewise function below of hotel rates per night, how much would it cost to stay 8 nights in the hotel?

\\f(x) = \

$$\begin{cases} 100x & 0 < x \\le 3 \\\\ 90x & 4 \\le x \\le 7 \\\\ 80x & x \\ge 8 \\end{cases}$$

\\

• $720
• $100
• $640
• $800

Explanation:

Identify the relevant interval for the input

The input value is \(x = 8\). We look at the domain intervals of the piecewise function \(F(x)\):

• \(0 < x \le 3\)
• \(4 \le x \le 7\)
• \(x \ge 8\)

Since \(8 \ge 8\), the input \(x = 8\) falls into the third interval.

Evaluate the function using the corresponding rule

For the interval \(x \ge 8\), the function rule is:

$$F(x) = 80x$$

Substitute \(x = 8\) into this expression:

$$F(8) = 80 \times 8 = 640$$

Match the result with the given choices

The calculated cost is \(\$640\), which corresponds to the third option.

Answer:

• (A) $720
• (B) $100
• (C) $640 (Correct answer)
• (D) $800