Composite Function For Discounts: Find The Final Sale Price

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Hey guys! Let's dive into a common scenario we encounter while shopping: discounts and coupons! Understanding how these apply and how to calculate the final price can be a real money-saver. In this article, we're going to break down how to choose the correct composite function to determine the final sale price after a 10% discount is applied, followed by a $150 coupon. This is a practical application of mathematical functions, and by the end, you'll be a pro at calculating the best deals!

Understanding Composite Functions

Before we jump into the specific problem, let's quickly recap what composite functions are. A composite function is basically a function that is applied to the result of another function. Think of it like a series of steps: you perform one action, and then you perform another action on the result of the first one. In mathematical terms, if you have two functions, let's say P(x) and C(x), the composite function P(C(x)) means you first apply the function C to x, and then you apply the function P to the result. The order is crucial here; P(C(x)) is generally not the same as C(P(x)). This concept is super important when we're dealing with discounts and coupons because the order in which they are applied affects the final price. Imagine getting a 10% discount first and then a $150 coupon versus the other way around – the final price will be different!

Discounts as Functions

In our scenario, we have a percentage discount and a fixed coupon amount. We can represent each of these as a function. Let's say x represents the original price of an item. A 10% discount means you're paying 90% of the original price, which can be written as the function P(x) = 0.9x. This function takes the original price x and returns the price after the 10% discount. Remember, understanding this conversion is key to setting up the problem correctly. The 0.9 represents the remaining percentage of the price after the 10% discount is taken off. So, if the original price is $100, then P(100) = 0.9 * 100 = $90. This makes intuitive sense; a 10% discount on $100 should leave you with $90 to pay.

Coupons as Functions

Now let's consider the coupon. A $150 coupon is a fixed amount that is subtracted from the price. We can represent this as the function C(x) = x - 150. This function takes the price x and subtracts $150 from it. Simple, right? If an item costs $500, then C(500) = 500 - 150 = $350. The critical part is realizing that we're not multiplying by a percentage here; we're subtracting a fixed value. This difference is important when we start combining these functions.

Determining the Correct Order of Operations

The core of the problem lies in figuring out the correct order in which to apply these functions. The question specifies that the 10% discount is applied first, followed by the $150 coupon. This is our roadmap for building the composite function. Since the discount comes first, we apply the P(x) function first. Then, we apply the coupon, which is the C(x) function, to the result of the discount. This means we're looking for the composite function C(P(x)). Think of it like this: we're taking the price after the discount and then applying the coupon. This order directly reflects the wording of the problem.

Why Order Matters

To really drive this point home, let's consider what would happen if we did it the other way around, i.e., P(C(x)). This would mean we apply the coupon first, subtracting $150 from the original price, and then apply the 10% discount. Mathematically, this would look like 0.9(x - 150). Notice how this is different from what we'll get with C(P(x)). The 10% discount is now being applied to a price that's already had $150 taken off. This can lead to a significantly different final price, especially for more expensive items. Understanding this difference is crucial for making informed shopping decisions and ensuring you're getting the best possible deal.

Constructing the Composite Function: C(P(x))

Okay, now let's put it all together. We know that P(x) = 0.9x (the 10% discount) and C(x) = x - 150 (the $150 coupon). We want to find C(P(x)), which means we need to substitute P(x) into the C(x) function. So, wherever we see x in C(x), we'll replace it with 0.9x. This gives us:

C(P(x)) = P(x) - 150

Now, substitute 0.9x for P(x):

C(P(x)) = 0.9x - 150

And there you have it! The composite function that represents a 10% discount followed by a $150 coupon is C(P(x)) = 0.9x - 150. This is the key equation that will help us calculate the final sale price.

Analyzing the Other Options

Let's quickly look at the other options provided in the question to understand why they are incorrect:

  • B. P(C(x)) = 0.9x - 150: As we discussed earlier, this represents applying the coupon before the discount, which is not what the question asked for. While the equation looks similar, the order of operations is different, making it incorrect.
  • C. C(P(x)) = 1.9x - 150: This is incorrect because it seems to be adding 1 to the discount percentage (0.9), which doesn't make sense in the context of the problem. There's no logical reason why we would multiply the original price by 1.9.
  • D. P(C(x)) = 1.9x - 150: This option combines both mistakes: applying the coupon first and using the incorrect multiplier of 1.9. It's definitely not the right choice.

Applying the Composite Function: An Example

To make this even clearer, let's use the example provided in the question: a laptop originally costs $800. We can use our composite function C(P(x)) = 0.9x - 150 to find the final sale price.

  1. Substitute the original price: x = 800
  2. Plug it into the function: C(P(800)) = 0.9(800) - 150
  3. Calculate:
    • 0.9 * 800 = 720
    • 720 - 150 = 570

So, the final sale price of the laptop after a 10% discount and a $150 coupon is $570. See how easy that was? Once you have the correct composite function, plugging in the values is straightforward.

Key Takeaways

Let's summarize the key things we've learned in this article:

  • Composite functions are functions applied in sequence, where the output of one function becomes the input of another.
  • The order of operations matters when dealing with discounts and coupons. Applying a discount before a coupon yields a different result than applying the coupon first.
  • A 10% discount can be represented as the function P(x) = 0.9x.
  • A $150 coupon can be represented as the function C(x) = x - 150.
  • To find the final price after a 10% discount followed by a $150 coupon, the correct composite function is C(P(x)) = 0.9x - 150.

Why This Matters in Real Life

Understanding composite functions and how they apply to discounts and coupons isn't just a math exercise; it's a practical skill that can save you money. By knowing how these things work, you can make informed decisions about which deals to take advantage of. Think about it: stores often use promotions that involve a combination of percentage discounts and fixed-amount coupons. Being able to quickly calculate the final price helps you compare different offers and choose the one that gives you the best bang for your buck. Plus, you'll avoid any surprises at the checkout counter!

Final Thoughts

So, there you have it! We've successfully navigated the world of composite functions and discounts. Remember, the key is to break down the problem into smaller steps, represent each step as a function, and then combine those functions in the correct order. With a little practice, you'll be a master of calculating sale prices in no time. Happy shopping, guys! And remember, always double-check your calculations to make sure you're getting the best possible deal. You've got this! Now go out there and snag some bargains using your newfound knowledge of composite functions. You'll be the savviest shopper around!