Lucy needs to buy some organic apples, and her grocery store is having a sale on them. If she buys 3 or fewer pounds of apples, the price will be $1.50 per pound. If she buys more than 3 pounds of apples, the price is $1.10 per pound. What is the domain of the piecewise-defined function, where [tex]$x$[/tex] represents the number of pounds of apples?
[tex] \{x \mid x \geq 0\} [/tex]
[tex] \{x \mid x \text{ is a real number }\} [/tex]
[tex] \{x \mid 0 \leq x \leq 3\} [/tex]
[tex] \{x \mid x \geq 3\} [/tex]
Answer (2)
The number of pounds of apples Lucy buys, x , must be non-negative.
There is no upper limit to the number of pounds of apples she can buy.
Therefore, the domain is all non-negative real numbers.
The domain of the piecewise-defined function is x ∣ x ≥ 0 .
Explanation
Understanding the Problem Let's analyze the problem. Lucy is buying apples, and the price per pound changes depending on how many pounds she buys. We need to find the domain of the function that represents this situation, where x is the number of pounds of apples.
Determining Possible Values of x The domain of a function is the set of all possible input values. In this case, the input is the number of pounds of apples, x . Since Lucy can't buy a negative amount of apples, x must be greater than or equal to 0. There's no limit to how many apples she can buy, so x can be any non-negative real number.
Expressing the Domain Therefore, the domain of the function is all real numbers greater than or equal to 0. In set notation, this is written as x ∥ xg e q 0 .
Final Answer The correct answer is x mi d xg e q 0 .
Examples
Understanding the domain of a function is crucial in many real-life scenarios. For instance, if you're calculating the profit of a lemonade stand based on the number of cups sold, the domain would be the number of cups you can sell, which can't be negative. Similarly, if you're modeling the height of a ball thrown in the air as a function of time, the domain would be the time from when the ball is thrown until it hits the ground. Knowing the domain helps you make sense of the function's output and avoid nonsensical results.
The domain of the piecewise-defined function, representing the pounds of apples Lucy can buy, is all non-negative real numbers. This is expressed as { x ∣ x ≥ 0 } . Therefore, the chosen option is oxed{\{x \mid x \geq 0\}} .
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