At a coffee shop, the menu consists of items and their prices. Is price a function of the item? Is the item a function of the price? Justify your answers.
Understand the Problem
The question is asking whether price is a function of the item on a coffee shop menu and whether the item can be considered a function of the price. It requires an explanation and justification for the answers given.
Answer
Price is a function of the item; the item is not a function of the price.
Answer for screen readers
- Price is a function of the item.
- The item is not a function of the price.
Steps to Solve
- Understanding Functions
A function is a relationship between two sets where every input is related to exactly one output. In this context, we analyze whether the price and item relationship meets this criterion.
- Is Price a Function of the Item?
When considering whether price is a function of the item, we can say that each item at the coffee shop corresponds to one particular price. For example, a latte priced at $3.50 does not change based on the quantity ordered. Thus, for each item, there is a unique price associated with it.
- Example of Price as a Function
If we denote the item as ( x ) and the price as ( y ), we can represent this relationship as:
$$ y = f(x) $$
where ( f ) is a function defining the price based on the item.
- Is the Item a Function of the Price?
Conversely, the question remains whether the item could be considered a function of the price. This relationship can become ambiguous, as multiple items could potentially have the same price. For example, both a cappuccino and a muffin might each cost $2.50. This means that the price does not uniquely determine one specific item.
- Example of Item Not being a Function of the Price
If we let ( y ) represent price and ( x ) represent the item, the relation cannot be expressed as:
$$ x = g(y) $$
because for a given price like $2.50, ( g(y) ) could yield multiple items (e.g., cappuccino and muffin).
- Price is a function of the item.
- The item is not a function of the price.
More Information
In a coffee shop, each menu item has a designated price, confirming that price can be viewed as a function of the item. However, since multiple items can share the same price, the item cannot be considered a function of the price.
Tips
- Confusing the relationship: A common mistake is to think that since prices often vary, they are not a function of items. In fact, each individual item does have a specific price.
- Misunderstanding the definition of a function: Forgetting that to qualify as a function, each input must correspond to only one output.
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