Question

1. at a convenience store, bottles of water cost $1.20 each. the function f(x) = 1.2x gives the cost of buying x bottles. give a reasonable domain and range for the function in this context.

Explanation:

Step1: Analyze Domain (x - number of bottles)

In the context of buying bottles, \( x \) represents the number of bottles. You can't buy a negative number of bottles, and typically, the number of bottles is a non - negative integer (you can buy 0, 1, 2, ... bottles). So the domain is the set of non - negative integers, i.e., \( x\in\{0, 1, 2, 3, \dots\} \) (or we can also consider practical upper limits, like the store's stock, but generally, we start with non - negative integers as a base).

Step2: Analyze Range (f(x) - cost)

The function is \( f(x)=1.2x \). Since \( x \) is a non - negative integer, when \( x = 0 \), \( f(0)=1.2\times0 = 0 \); when \( x = 1 \), \( f(1)=1.2\times1=1.2 \); when \( x = 2 \), \( f(2)=1.2\times2 = 2.4 \); and so on. So the range is the set of values \( f(x)=1.2x \) where \( x \) is a non - negative integer. In other words, the range is \( \{0, 1.2, 2.4, 3.6, \dots\} \), or we can express it as the set of all non - negative multiples of 1.2 (where the multiplier is a non - negative integer). If we consider a more practical situation (e.g., the store has a limited number of bottles, say up to \( N \) bottles), the range would be \( \{0, 1.2, 2.4, \dots, 1.2N\} \), but without a specific upper limit on stock, we use the set of non - negative multiples of 1.2 with \( x\in\mathbb{Z}_{\geq0} \).

Answer:

Domain: The set of non - negative integers (\( x = 0,1,2,3,\dots \))
Range: The set of non - negative multiples of 1.2 (\( f(x)=0, 1.2, 2.4, 3.6,\dots \)) (or, if we consider a practical context with a maximum number of bottles, the range is values of the form \( 1.2x \) where \( x \) is a non - negative integer within the practical limit of the number of bottles available for purchase).