Multiply Linear Functions: F(x) And G(x) Explained

Dec 16, 2025 by Andrew McMorgan 51 views

Hey guys! Today, we're diving into the awesome world of linear functions and tackling a super common problem: multiplying them. You know, those functions like $f(x)=x-2$ and $g(x)=-3x+4$ ? We're going to figure out what happens when you multiply them together, or in math terms, determine $(f imes g)(x)$ . This isn't just some abstract concept; understanding how to multiply functions is a key skill in algebra and pops up in all sorts of places, from calculus to physics. So, grab your notebooks, maybe a snack, and let's break down how to find the product of $f(x)$ and $g(x)$ . We'll go through it step-by-step, making sure you guys can follow along and feel confident about this. By the end, you'll see how that multiplication leads to a brand new function, and we'll even look at the options to pinpoint the correct answer. Get ready to flex those algebraic muscles!

Understanding Function Multiplication

Alright, let's get down to business. When we talk about determining $(f imes g)(x)$ , we're essentially saying we want to find the product of the two functions, $f(x)$ and $g(x)$ . Think of it like this: if $f(x)$ represents one set of instructions and $g(x)$ represents another, $(f imes g)(x)$ is the result of performing those two sets of instructions simultaneously through multiplication. For our specific problem, we have $f(x) = x-2$ and $g(x) = -3x+4$ . To find $(f imes g)(x)$ , we simply need to multiply the expression for $f(x)$ by the expression for $g(x)$ . So, we're looking at $(x-2) imes (-3x+4)$ . This looks like a pretty standard multiplication problem, right? We've probably all done this with polynomials before. The key here is to use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last) when dealing with two binomials. It's a systematic way to make sure every term in the first expression gets multiplied by every term in the second expression. It’s crucial to get this part right, as a single mistake in distribution can lead you to the wrong final answer, and trust me, we don't want that. We want to be accurate and nail this problem. So, let's visualize this. The 'First' terms are $x$ and $-3x$ . The 'Outer' terms are $x$ and $4$ . The 'Inner' terms are $-2$ and $-3x$ . And finally, the 'Last' terms are $-2$ and $4$ . By multiplying these pairs, we get the terms of our new, expanded function. This process might seem straightforward, but it requires careful attention to detail, especially with the signs. A negative multiplied by a positive is negative, and a negative multiplied by a negative is positive. Keeping track of these sign rules is absolutely essential for success. We'll go through each of these multiplications explicitly in the next section, showing you exactly how the resulting terms come together. This foundational understanding of function multiplication and the distributive property is what allows us to conquer problems like this one with confidence. It’s the bedrock upon which more complex algebraic manipulations are built, so let's make sure we’ve got it down pat.

Performing the Multiplication: Step-by-Step

Now that we understand what we're doing, let's actually do it. We need to calculate $(f imes g)(x)$ where $f(x)=x-2$ and $g(x)=-3x+4$ . As we discussed, this means we're multiplying the two expressions: $(x-2)(-3x+4)$ . Let's use the FOIL method to make sure we cover all our bases. Remember, FOIL stands for First, Outer, Inner, Last.

First: Multiply the first terms in each binomial. That's $x$ from $(x-2)$ and $-3x$ from $(-3x+4)$ . So, $x imes (-3x) = -3x^2$ . This is our first term in the resulting polynomial.
Outer: Multiply the outer terms. That's $x$ (the first term of the first binomial) and $4$ (the last term of the second binomial). So, $x imes 4 = 4x$ . This is our second term.
Inner: Multiply the inner terms. That's $-2$ (the last term of the first binomial) and $-3x$ (the first term of the second binomial). So, $(-2) imes (-3x) = 6x$ . Remember, a negative times a negative is a positive! This is our third term.
Last: Multiply the last terms in each binomial. That's $-2$ from $(x-2)$ and $4$ from $(-3x+4)$ . So, $(-2) imes 4 = -8$ . This is our final term.

Now, we put all these resulting terms together: $-3x^2 + 4x + 6x - 8$ . Looking at this expression, we can see that we have two 'like terms' – terms that have the same variable raised to the same power. These are $4x$ and $6x$ . We can combine these like terms by adding their coefficients. So, $4x + 6x = 10x$ .

Therefore, our final simplified expression for $(f imes g)(x)$ is $-3x^2 + 10x - 8$ . This process is absolutely fundamental. It’s the application of the distributive property that allows us to expand and simplify the product of two algebraic expressions. Each step is critical: correctly identifying the pairs to multiply, performing the multiplication accurately (paying close attention to signs), and then combining any like terms to achieve the most simplified form. It’s not just about getting the answer; it’s about mastering the technique. This method ensures that no part of either function is left out of the multiplication, resulting in a complete and accurate product. So, when you see problems like this, just break it down, use FOIL (or another systematic distribution method), and remember to combine those like terms. It's as simple as that, and you'll be multiplying functions like a pro!

Analyzing the Options

So, we've done the math, and we've arrived at our answer: $(f imes g)(x) = -3x^2 + 10x - 8$ . Now, let's take a look at the multiple-choice options provided to see which one matches our result. This is a crucial step in any test or quiz situation, ensuring our calculations align with the expected format of the answer.

A. $(f imes g)(x) = -3x - 8$ B. $(f imes g)(x) = -3x^2 - 8$ C. $(f imes g)(x) = -3x^2 + 10x - 8$ D. $(f imes g)(x) = -3x^2 - 2x - 8$

Let's compare our calculated result, $-3x^2 + 10x - 8$ , with each option:

Option A is $-3x - 8$ . This option is missing the $x^2$ term and has the wrong coefficient for the $x$ term. It doesn't match.
Option B is $-3x^2 - 8$ . This option has the correct $x^2$ term and the constant term, but it's missing the $x$ term entirely. It doesn't match.
Option C is $-3x^2 + 10x - 8$ . This option matches our calculated result exactly! It has the $-3x^2$ from the 'First' multiplication, the $+10x$ from combining the 'Outer' ( $4x$ ) and 'Inner' ( $6x$ ) terms, and the $-8$ from the 'Last' multiplication. Bingo!
Option D is $-3x^2 - 2x - 8$ . This option has the correct $x^2$ term and constant term, but the coefficient of the $x$ term is incorrect. It seems like a possible error in combining the outer and inner terms, perhaps mishandling the signs or coefficients. It doesn't match.

Therefore, based on our detailed step-by-step multiplication and comparison, the correct answer is Option C. It's super satisfying when your calculated answer lines up perfectly with one of the choices, right? It confirms that our process was sound and our understanding of multiplying linear functions was spot on. Always double-check your work and compare it carefully against the given options. This systematic approach helps eliminate errors and build confidence in your mathematical abilities. Remember, every part of the calculation matters, from the initial distribution to the final simplification. Getting the right answer is a testament to paying attention to every detail.

Why This Matters: Beyond the Classroom

So, why do we even bother learning how to multiply functions like $f(x)=x-2$ and $g(x)=-3x+4$ ? Is this just something confined to textbooks and math exams, guys? Absolutely not! Understanding function multiplication is a fundamental building block in mathematics with far-reaching applications. When you multiply functions, you're essentially creating a new function that combines the behaviors or outputs of the original two. This concept is crucial in fields like economics, where you might model revenue as the product of price per unit and the number of units sold. For example, if $p(x)$ is the price of a product when $x$ units are demanded, and $q(x)$ is the quantity demanded, then the total revenue $R(x)$ could be modeled as $R(x) = p(x) imes q(x)$ . Analyzing this new revenue function ( $R(x)$ ) helps businesses understand profitability, predict sales, and make strategic decisions. In physics, especially in areas like signal processing or wave mechanics, the product of two functions can represent interference patterns or the modulation of one signal by another. Imagine multiplying a carrier wave by a modulating signal to transmit information – the resulting function contains information about both original signals. Even in computer graphics, complex transformations and effects are often achieved by multiplying various transformation matrices (which can be thought of as functions). Scaling, rotation, and translation operations are applied sequentially, and their combined effect is represented by the product of these individual transformations. Furthermore, in calculus, the product rule for differentiation, which deals with the derivative of a product of two functions, directly relies on the concept of function multiplication. Understanding $(f imes g)(x)$ is a prerequisite for mastering differentiation and integration techniques that are essential for solving complex problems in science and engineering. So, while the specific problem of multiplying $f(x)=x-2$ and $g(x)=-3x+4$ might seem basic, the underlying skill of function multiplication is a powerful tool. It allows us to model more complex real-world scenarios by combining simpler functions, leading to deeper insights and the ability to solve more sophisticated problems across a wide range of disciplines. It’s a key piece of the mathematical puzzle that unlocks a deeper understanding of how different mathematical relationships can be combined and analyzed. Keep practicing, and you'll see these concepts pop up in more places than you might expect!