Question

suppose kayla is hosting a party and knows at least 6 are coming. the party is capped at 16 guests. let $g(x)$ model the number of tables kayla needs to set up if $x$ guests attend. what is the domain of the function? use set notation.

guests	tables needed
10 - 13	2
14 - 16	3

Explanation:

Step1: Identify the range of guests

The problem states at least 6 guests are coming and the party is capped at 16 guests. Also, from the table, the guest ranges are 6 - 9, 10 - 13, and 14 - 16. So \( x \) (number of guests) must be an integer (since you can't have a fraction of a guest) from 6 to 16 inclusive.

Step2: Write the domain in set notation

The domain is the set of all possible \( x \) values. So \( x \) is an integer where \( 6\leq x\leq16 \). In set notation, this is \( \{x\in\mathbb{Z}\mid 6\leq x\leq16\} \), where \( \mathbb{Z} \) represents the set of integers. But since the guest counts are whole numbers (you can't have a fraction of a guest), and the ranges are 6 - 9, 10 - 13, 14 - 16, we can also list the possible integer values from 6 to 16. So the domain is the set of integers from 6 to 16, inclusive.

Answer:

\(\{x \in \mathbb{Z} \mid 6 \leq x \leq 16\}\) (or equivalently \(\{6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\) since \(x\) must be an integer representing the number of guests)