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Let’s walk through some application problems for working with functions.
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Here’s an example.
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Mark wants to calculate the total cost for reserving a number of nights at a particular hotel.
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If the cost for a night is fifty dollars plus a registration fee of twenty-five dollars, write a function that describes the total cost based on the number of nights.
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What is the initial cost?
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First, we wanna make sure we know what the problem is asking of us.
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It’s asking that we write a function.
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It’s also asking a question, what is the initial cost.
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To answer these questions, let’s highlight all the information we were given.
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The cost of the hotel for a night is fifty dollars.
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There’s a registration fee of twenty-five dollars.
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And we’re trying to write a function that describes the total cost based on the number of nights.
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We know that a function is a relationship between numbers.
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It’s a specific relationship where every input has exactly one output.
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In our case, we need to input the number of nights.
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And we need the output to be the total cost.
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Before we move any further with our function, let’s answer the question, what is the initial cost.
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The initial cost is twenty-five dollars; it’s the registration fee for the hotel.
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You pay the initial cost or the registration fee one time, no matter how many nights you stay.
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It’s your initial cost for the hotel.
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Let’s set up a table.
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For one night, you would pay fifty dollars for the room plus twenty-five dollars for the registration fee.
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For the second night, you would pay one hundred dollars for the room plus twenty-five dollars for the registration fee.
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We know that the number of nights is our 𝑥-value and the cost is our 𝑦-value.
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So our function, 𝑦 equals the cost per night plus the initial cost.
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In this case, it’s fifty dollars times 𝑥 plus twenty-five.
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Fifty times 𝑥, where 𝑥 is the number of nights and twenty-five dollars is the registration fee.
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We’ll just clear up the screen a little bit, and then check and see, have we answered both of the things the question was asking us.
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Did we write a function?
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Yes.
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This is a function that describes the total cost based on the number of nights.
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And we answered the question, what was the initial cost, which is twenty-five dollars for the registration fee.
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Here’s another example.
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An artist bought some paintbrushes for five dollars each.
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He had a coupon for two dollars and thirty-one cents off his purchase.
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Write a function to represent the total purchase price.
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Let’s check what the problem is asking of us.
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It wants us to write a function.
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We need to find the information we were given.
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Here’s the information we’re given.
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He bought some paintbrushes, we don’t know how many.
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They cost five dollars each.
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He had a coupon for two dollars and thirty-one cents off.
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And then we want our function to represent the total purchase price.
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So we’ll have an 𝑥-value going in and some 𝑦-value going out.
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This is the variable that the price depends on, the number of paintbrushes will tell us how much the total price should be.
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We’re going to label them with 𝑥 and 𝑦 for the input and the output.
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Inside our function rule box, something is going to happen.
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We know that each of our 𝑥-values cost five dollars.
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That means that for every paintbrush you buy, you pay five dollars.
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This is represented by multiplication, more simply five 𝑥.
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But, does five dollars times 𝑥 equal the total purchase price.
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It would if he didn’t have a coupon, but he does.
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This means that no matter what he buys, the artist will take two dollars and thirty-one cents off the price of the paintbrushes.
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The operation that represents that is subtraction.
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We’ll take away two dollars and thirty-one cents.
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And this is our function, 𝑦 equals five 𝑥 minus two dollars and thirty-one cents.
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We could use this function to find the total purchase price, if the artist bought one paintbrush, if the artist bought seventeen paintbrushes.
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It wouldn’t matter; this function would still work.
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Here’s our last example.
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Find the domain and range of the following function: 𝑦 equals three 𝑥 and 𝑥 is seven, eight, nine, and ten.
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Let’s pause the question here and quickly review what domain and range mean.
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Domain is the set of all the input values and range is the set of all the output values.
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So in our function, domain, the set of all the input values together, range, the set of all the output values.
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Remember, we usually use 𝑥 and 𝑦 to represent our function.
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So another way to describe the domain would be the set of all the 𝑥-values.
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And another way to describe the range would be the set of all the 𝑦-values.
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Back to the problem on hand, we can make a table to find the domain and the range.
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We were already given the 𝑥-values, which means we were already given the domain.
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Now we need to use our function to calculate the range.
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So we need to solve three times seven equals twenty-one.
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Three times eight is twenty-four, three times nine is twenty-seven, three times ten is thirty.
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And these values make up the range.
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And if we write them out, they look like this.
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The domain, seven, eight, nine, ten.
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The range, twenty-one, twenty-four, twenty-seven, and thirty.
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Those were only a few applications of functions.
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You’ll find many more as you explore and practice.