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Recognizing functions from verbal description word problems

Checking whether a description of the price of an order can be represented as a function of the shipping cost. Created by Sal Khan.

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Video transcript

Jada is ordering Mother's Day gifts online. The shipping costs are based on the dollar amount of the order. For orders less than $20, shipping costs $4. For orders $20 or more, shipping is $7. Can the dollar amount of the order be represented as a function of shipping costs? So they're saying, can the dollar amount of the order-- so can the amount of order be represented as a function of shipping-- let me do that in that blue color costs. So if we have the shipping costs as an input, will that map to exactly-- for a given input, will we get exactly one output for the amount of order? In order for this to be represented as a function, we have to input a shipping cost, a shipping cost where this relationship is defined. We need to input a shipping cost, put it into our relationship, and get exactly one dollar amount of the order in order for this to be a function. If we get multiple dollar amounts of the order, then the relationship, well, it's still a relationship, but it's not going to be a function. So let's think about it. What are the possible inputs here? Well, there's only two possible shipping costs. Shipping costs are either going to be $4, or they're going to be $7. So let's think about what happens when we input $4 in as a shipping cost. So if we input $4 into our relationship, so we input $4 into our little potential function box, so $4 into it, what is the output? What is going to be the amount of the order? Well, if the shipping cost is $4, the amount of the order just has to be anything less than $20. So it could have been $1. It could have been $1.50. It could've been $7. It literally can take on any value up to $20. So it could even be $19.99. We could do a similar thing if we input 7 into this relationship. If 7 was-- and I could put literally an infinite number of numbers. It could be a million dollars. So if I input 7 into this relationship that we're trying to test whether it's a function, if 7 is the shipping costs, then the order we just know is over $20, $20 or more. So it could be $20. It could be $800. It could be $1 million. There's actually an infinite number of values that it could take on right over here. So because for a valid shipping cost, for each of these valid shipping costs, I can get many, many, many potential outputs, I don't know which output it necessarily will output to. If someone tells you the shipping costs and you don't necessarily know what the order size was, this is not a function. You cannot represent the amount of order as a function of the shipping cost. So, no.