Question

use the graph of the piecewise function to answer the questions in context.

1. generic strawberry fields allows customers to pay $5 to pick strawberries plus 50 cents for every pint or partial pint of strawberries that they pick. there is a limit of 8 pints per customer. the piecewise function f shown below models the price of strawberries picked.

a. what is the domain in this context?
b. the range can be represented by ( y = 5+0.5x ) where x is all integers in the domain. explain why.
c. find ( f(3.5) ). what does it mean in this context?

Explanation:

a. The domain represents the possible values of the number of pints of strawberries picked. Since a customer can pick more than \(0\) pints (because you can't pick a negative or zero amount if you are paying for the service, and there is a limit of \(8\) pints per customer), so the domain is \(0\lt x\leq8\) where \(x\) is a non - negative real number (because you can pick a partial pint).
b. The formula \(y = 5 + 0.5x\) is a linear formula. The \(5\) is the fixed cost (entrance fee to pick strawberries), and \(0.5x\) is the variable cost (cost per pint). Although \(x\) in the domain can be a non - integer (for partial pints), when calculating the range values (the cost), we use the fact that for any non - integer \(x\) in the domain, the cost is calculated based on the next - higher integer value of \(x\) (as per the piece - wise function's construction, which is similar to a ceiling function in a sense, but when using the formula \(y=5 + 0.5x\) with \(x\) as integers in the domain, we can generate all the range values).
c. Looking at the piece - wise function graph, when \(x = 3.5\), we find the corresponding \(y\) - value. Since \(3\lt3.5\leq4\), we use the formula or the graph's step for \(x = 4\) (because of how the piece - wise function is defined for partial pints). Substituting \(x = 4\) into \(y=5 + 0.5x\) (or reading from the graph), \(y=5+0.5\times4=5 + 2=7\). So \(f(3.5) = 7\), which is the cost for picking \(3.5\) pints of strawberries.

Answer:

a. \(0\lt x\leq8\)
b. The cost function \(y = 5+0.5x\) is used because the base cost is \(5\) dollars and for each integer - valued number of pints \(x\) (since the price is charged per whole or partial pint, but when calculating the range values based on the domain of non - negative real numbers up to \(8\), the range values are calculated using integer increments of \(x\) for the formula), we add \(0.5x\) to the base cost.
c. \(f(3.5)=7\). In this context, it means that when a customer picks \(3.5\) pints of strawberries, the cost of the strawberries is \(7\) dollars.