Combining Functions: Algebra And Demand Analysis

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically focusing on how we can combine functions and analyze real-world scenarios using them. We'll tackle some algebraic function operations and then switch gears to look at a practical example involving product demand and pricing. So, buckle up, grab your calculators, and let's get mathematical!

Understanding Function Operations: Addition and Subtraction

First up, let's talk about combining functions through addition and subtraction. It's like mixing ingredients in a recipe; you're putting two things together to create something new. We've got two functions here to play with: f(x)=rac{5}{3} x^2+2 x-rac{5}{8} and $g(x)=3 x^2$. Our mission, should we choose to accept it, is to find the rules for the new functions that result from adding and subtracting these two.

The Rule for $f+g$

Alright, let's tackle question 20: finding the rule for $(f+g)(x)$. This notation simply means we need to add the functions $f(x)$ and $g(x)$ together. Think of it as taking all the terms from $f(x)$ and all the terms from $g(x)$ and putting them into one big happy function. So, we'll write out $f(x)$ and $g(x)$ and then combine the like terms. Remember, like terms are terms that have the same variable raised to the same power. In our case, the terms with $x^2$ are like terms, and the terms with just $x$ and the constant terms are also unique categories. Let's get started. We have f(x) = rac{5}{3} x^2 + 2x - rac{5}{8} and $g(x) = 3x^2$. To find $(f+g)(x)$, we simply add them:

$(f+g)(x) = f(x) + g(x)$

(f+g)(x) = igg(rac{5}{3} x^2 + 2x - rac{5}{8}igg) + (3x^2)

Now, we group the like terms. The $x^2$ terms are rac{5}{3} x^2 and $3x^2$. To add these, we need a common denominator. The common denominator for 3 and 1 (which is the denominator of 3) is 3. So, we can rewrite $3x^2$ as rac{9}{3} x^2. Now, adding the coefficients:

rac{5}{3} x^2 + rac{9}{3} x^2 = rac{5+9}{3} x^2 = rac{14}{3} x^2

The term $2x$ has no other like terms in $g(x)$, so it remains $2x$. The constant term -rac{5}{8} also has no like terms in $g(x)$, so it stays -rac{5}{8}. Putting it all together, the rule for $(f+g)(x)$ is:

(f+g)(x) = rac{14}{3} x^2 + 2x - rac{5}{8}

See? Not too shabby! We just combined the polynomial parts. The key here is to identify and combine those similar terms. This operation is fundamental in algebra, and understanding it helps build a strong foundation for more complex mathematical concepts. When you add functions, you're essentially creating a new function whose output is the sum of the outputs of the original functions for any given input. This can be useful in modeling scenarios where two different effects are cumulative. For instance, if $f(x)$ represented the cost of producing a certain item and $g(x)$ represented the overhead costs, then $(f+g)(x)$ would represent the total cost.

The Rule for $f-g$

Next up, let's conquer question 21: finding the rule for $(f-g)(x)$. This is very similar to addition, but with a crucial difference – we're subtracting $g(x)$ from $f(x)$. When you subtract a function, you're essentially distributing the negative sign to every term in the function being subtracted. This is a common place where mistakes can happen, so pay close attention! We start with the same functions: f(x) = rac{5}{3} x^2 + 2x - rac{5}{8} and $g(x) = 3x^2$. Now, let's subtract:

$(f-g)(x) = f(x) - g(x)$

(f-g)(x) = igg(rac{5}{3} x^2 + 2x - rac{5}{8}igg) - (3x^2)

Now, we distribute the negative sign to each term inside the second parenthesis:

(f-g)(x) = rac{5}{3} x^2 + 2x - rac{5}{8} - 3x^2

Just like before, we identify and combine the like terms. The $x^2$ terms are rac{5}{3} x^2 and $-3x^2$. We need that common denominator again. Rewriting $3x^2$ as rac{9}{3} x^2, our subtraction becomes:

rac{5}{3} x^2 - rac{9}{3} x^2 = rac{5-9}{3} x^2 = rac{-4}{3} x^2

Since we're subtracting $g(x)$, and $g(x)$ only had an $x^2$ term, the $2x$ term and the -rac{5}{8} term from $f(x)$ remain unchanged. So, combining everything, the rule for $(f-g)(x)$ is:

(f-g)(x) = -rac{4}{3} x^2 + 2x - rac{5}{8}

And there you have it! The subtraction of functions also results in a new function. This operation is incredibly useful when you want to find the difference between two quantities, like profit (revenue minus cost) or net change. In our example, if $f(x)$ represented one type of revenue stream and $g(x)$ represented another, $(f-g)(x)$ could help analyze the difference in their contributions. Always remember to be careful with the signs when subtracting functions, guys. It's a small detail that can make a big difference in your final answer.

Real-World Applications: Demand and Price Analysis

Now, let's shift gears from abstract algebra to a more tangible application. Question 22 introduces a scenario involving a company's jeans. We're given a demand function, $d(x) = 6,500 - 6.83x$, where $d$ represents the number of units sold (demand) and $x$ represents the price of the jeans in dollars. This type of function is super common in economics and business, as it helps companies understand how changes in price affect how much of their product people will buy. It's a linear function, meaning the relationship between price and demand is a straight line.

Understanding the Demand Function

Let's break down $d(x) = 6,500 - 6.83x$. The number $6,500$ is the y-intercept. In this context, it represents the theoretical demand if the price were zero dollars. While a price of zero is unrealistic, it gives us a baseline. The number $-6.83$ is the slope of the line. This is the crucial part for understanding the relationship between price and demand. The negative sign tells us that as the price ($x$) increases, the demand ($d(x)$) decreases. Specifically, for every one-dollar increase in the price of the jeans, the company can expect to sell approximately 6.83 fewer pairs of jeans. This inverse relationship is a fundamental principle in economics – the law of demand.

Imagine you're the business owner. You want to set a price that maximizes your revenue, which is price multiplied by demand. If you set the price too high, you might make a lot of money per pair, but you'll sell very few. If you set the price too low, you'll sell a lot of pairs, but the revenue per pair will be small. Finding the sweet spot is key, and understanding your demand function is the first step.

Analyzing Demand Scenarios

The question poses a scenario: