Problem 101 - Precalculus Essentials

Question

The regular price of a computer is \(x\) dollars. Let \(f(x)=x-400\) and \(g(x)=0.75 x\) a. Describe what the functions \(f\) and \(g\) model in terms of the price of the computer. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the computer. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the computer, \(f^{\circ}\) g or \(g \circ f\) ? Explain.

Step-by-Step Solution

Answer

Function \(f\) models a flat discount of $400, while \(g\) models a 25% discount from the original price. \((f \circ g)(x) = 0.75x - 400\) models first applying a 25% discount, then subtracting $400. \((g \circ f)(x) = 0.75x - 300\) models subtracting $400 first and then applying a 25% discount. The greater discount comes from the \((f \circ g)(x)\), where a 25% discount is applied first followed by a flat $400 discount.

Step 1: Interpret Function \(f(x)\) and \(g(x)\)

\(\(f(x)=x-400\) models a flat discount of $400 on the original price. On the other hand, \(g(x)=0.75x\) models a discount of 25%, or in other words, the final price is 75% of the original price.

Step 2: Compute \((f \circ g)(x)\)

\((f \circ g)(x) = f(g(x)) = f(0.75x) = 0.75x - 400\). This models first applying the 25% discount and then subtracting a flat $400 from the discounted price.

Step 3: Compute \((g \circ f)(x)\)

\((g \circ f)(x) = g(f(x)) = g(x - 400) = 0.75(x - 400) = 0.75x - 300\). This models first subtracting a flat $400 from the original price, and then taking applying the 25% discount to the result.

Step 4: Determine the Greater Discount

Since \((f \circ g)(x) = 0.75x - 400\) reduces the price more than \((g \circ f)(x) = 0.75x - 300\), \((f \circ g)(x)\) offers the greater discount.

Key Concepts

Function Composition

Understanding function composition is like learning how to combine two different recipes to create a new dish. Just as recipes are followed in a certain order to cook a meal, functions can be combined in a particular sequence to form a composite function. This process involves taking the output from one function and using it as the input for another function. In mathematical terms, if we have two functions, say, f(x) and g(x), then the composite function (f \(\circ\) g)(x) is found by plugging g(x) into the function f(x), resulting in f(g(x)).
It's crucial to remember the order matters! The composite (f \(\circ\) g)(x) can yield a completely different result than (g \(\circ\) f)(x). Think of it like baking a cake before or after you've mixed the ingredients, the sequence affects the final outcome.

Discount Modeling

When addressing the topic of discount modeling, we're essentially examining how different discount methods affect the price of an item. In our exercise, two types of discounts are being modeled by functions: a flat discount and a percentage discount. The function f(x)=x-400 represents a flat discount where no matter the original price x, the discount is always $400. On the other hand, g(x)=0.75x models a percentage discount, which provides a price that is 75% of the original; this reflects a 25% reduction in price. Discount modeling is a practical application of functions in real life, particularly in retail and sales industries, where it's used to calculate final prices after discounts are applied.

Function Interpretation

Interpreting functions means understanding what they stand for and how they transform inputs into outputs. In our context, interpreting the functions f(x) and g(x) helps us see how each discount changes the price of the computer. By breaking down the functions, we learn that f(x) gives us a simple subtraction from the original price, while g(x) illustrates a proportional change. This distinction is critical when we need to explain or predict how different pricing strategies will affect the cost of an item. It's like being able to read a map; once you understand the symbols (functions) and the landscape (mathematical context), you can navigate the pricing world effectively.

Function Application

The application of functions to problems involves substituting specific values into the function or, in the case of our exercise, combining different functions to explore various scenarios. The application of (f \(\circ\) g)(x) and (g \(\circ\) f)(x) shows us two different strategies for applying discounts. We can see that the order of applying these functions affects the final discounted price. This tells us that we need to consider the sequence in which sales promotions are applied, as the final outcome is sensitive to this order. The ability to apply functions correctly is a powerful tool in financial planning, budgeting, and strategic decision-making in business practices.