Finding F(x)g(x) When F(x) Equals 2x-3 And G(x) Equals X^2+1

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Function multiplication is a fundamental operation in mathematics, especially in algebra and calculus. It involves combining two functions, f(x) and g(x), to create a new function. The process is straightforward: you simply multiply the expressions that define each function. In this comprehensive guide, we will explore how to find the product of two functions, specifically when f(x) = 2x - 3 and g(x) = x^2 + 1. This article aims to provide a clear, step-by-step explanation, ensuring readers of all backgrounds can grasp the concept and apply it effectively. We'll delve into the mechanics of polynomial multiplication, the importance of the distributive property, and the simplification of the resulting expression. Whether you're a student learning about functions for the first time or someone looking to refresh your algebra skills, this article will provide you with the knowledge and confidence to tackle similar problems. The beauty of mathematics lies in its precision and logical structure, and function multiplication is a prime example of this. By understanding the underlying principles and practicing the techniques, you can unlock a deeper appreciation for the world of functions and their applications.

Defining the Functions

Before we dive into the multiplication, let's clearly define the functions we're working with. We are given two functions: f(x) = 2x - 3 and g(x) = x^2 + 1. The function f(x) is a linear function, which means its graph would be a straight line. It takes an input x, multiplies it by 2, and then subtracts 3. The function g(x), on the other hand, is a quadratic function, meaning its graph would be a parabola. It takes an input x, squares it, and then adds 1. Understanding the nature of these functions is crucial for visualizing their behavior and how they interact when multiplied. Linear functions like f(x) have a constant rate of change, represented by the coefficient of x (in this case, 2). Quadratic functions like g(x) have a varying rate of change, dictated by the squared term. When we multiply these two functions, we are essentially creating a new function that combines the characteristics of both. This new function will be a cubic function, as the highest power of x will be 3. This is because we will be multiplying a term with x to the power of 1 (from f(x)) with a term with x to the power of 2 (from g(x)), resulting in a term with x to the power of 3. The process of multiplying these functions involves applying the distributive property, a fundamental concept in algebra. This property allows us to multiply each term in one expression by each term in the other expression, ensuring we account for all possible combinations. The resulting expression will then need to be simplified by combining like terms, a process that involves adding the coefficients of terms with the same power of x. This is a critical step in obtaining the final, simplified form of the multiplied function.

The Multiplication Process: Step-by-Step

The multiplication of f(x) and g(x), denoted as f(x)g(x), involves multiplying the expressions representing each function. Given f(x) = 2x - 3 and g(x) = x^2 + 1, we need to multiply (2x - 3) by (x^2 + 1). This process utilizes the distributive property, which states that a(b + c) = ab + ac. In our case, we will distribute each term in the first expression (2x - 3) over the terms in the second expression (x^2 + 1). First, we multiply 2x by both x^2 and 1: 2x * x^2 = 2x^3 and 2x * 1 = 2x. Next, we multiply -3 by both x^2 and 1: -3 * x^2 = -3x^2 and -3 * 1 = -3. After performing these multiplications, we obtain the following expression: 2x^3 + 2x - 3x^2 - 3. This expression contains four terms, each resulting from the multiplication of one term from f(x) and one term from g(x). However, this expression is not yet in its simplest form. To simplify it, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, there are no like terms, as each term has a different power of x. Therefore, the next step is to rearrange the terms in descending order of their powers of x. This is a standard practice in algebra that makes the expression easier to read and understand. By arranging the terms, we obtain the final expression for f(x)g(x), which is 2x^3 - 3x^2 + 2x - 3. This cubic function represents the product of the linear function f(x) and the quadratic function g(x). It is important to note that the order of multiplication does not matter, meaning f(x)g(x) is the same as g(x)f(x). This is due to the commutative property of multiplication, which holds for functions as well as numbers.

Simplifying the Result

After performing the multiplication, we arrive at the expression 2x^3 + 2x - 3x^2 - 3. The next crucial step is simplifying this expression. Simplification in algebra involves combining like terms and arranging the terms in a standard order, typically in descending order of exponents. In our expression, we have four terms: 2x^3, 2x, -3x^2, and -3. To identify like terms, we look for terms that have the same variable raised to the same power. In this case, there are no like terms, as each term has a unique power of x. The term 2x^3 has x raised to the power of 3, the term -3x^2 has x raised to the power of 2, the term 2x has x raised to the power of 1, and the term -3 is a constant term with x effectively raised to the power of 0. Since there are no like terms to combine, our focus shifts to arranging the terms in descending order of exponents. This means we want to place the term with the highest power of x first, followed by the term with the next highest power, and so on, until we reach the constant term. This standard form makes the polynomial easier to read and analyze. Following this order, we rearrange the terms as follows: 2x^3 - 3x^2 + 2x - 3. This is the simplified form of f(x)g(x). It is a cubic polynomial, which is a polynomial of degree 3. The coefficient of the x^3 term, which is 2, is called the leading coefficient. The constant term, which is -3, is the value of the polynomial when x is equal to 0. This simplified expression represents the product of the functions f(x) and g(x) and is the final result of our calculation. It is important to double-check the simplification process to ensure accuracy, as any errors in combining like terms or rearranging the terms can lead to an incorrect final answer.

Final Result: f(x)g(x) = 2x^3 - 3x^2 + 2x - 3

In conclusion, after meticulously performing the multiplication and simplification steps, we arrive at the final result for f(x)g(x). Given f(x) = 2x - 3 and g(x) = x^2 + 1, the product f(x)g(x) is equal to 2x^3 - 3x^2 + 2x - 3. This final expression represents a cubic function, which is a polynomial function of degree 3. The process involved applying the distributive property to multiply the two functions, followed by simplifying the resulting expression by combining like terms and arranging the terms in descending order of exponents. The result, 2x^3 - 3x^2 + 2x - 3, provides valuable insights into the behavior of the combined function. The leading term, 2x^3, dominates the function's behavior for large values of x, while the constant term, -3, represents the y-intercept of the function's graph. The other terms, -3x^2 and 2x, contribute to the function's shape and curvature. Understanding how to multiply functions and simplify the result is a fundamental skill in algebra and calculus. It allows us to combine functions in meaningful ways and analyze the resulting expressions. This process is widely used in various mathematical applications, including modeling real-world phenomena, solving equations, and optimizing functions. The steps we followed in this article provide a clear and systematic approach to function multiplication, ensuring accuracy and understanding. By mastering this technique, you can confidently tackle more complex mathematical problems involving functions and their operations. The final result, f(x)g(x) = 2x^3 - 3x^2 + 2x - 3, is a testament to the power of algebraic manipulation and the beauty of mathematical precision.