Problem 48 - Algebra 2

Question

Let \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2 .\) Perform each function operation. $$ -2 g(x)+f(x) $$

Step-by-Step Solution

Answer

The result of the function operation \(-2g(x)+f(x)\) is \(-2x^{2}+8x+1\).

Step 1: Identify Functions

The given functions are: \(f(x)=2 x+5\) and \(g(x)=x^{2}-3 x+2\). The operation that needs to be performed is \(-2g(x)+f(x)\).

Step 2: Perform Multiplication

The term \(-2g(x)\) means that each part of the function \(g(x)\) needs to be multiplied by \(-2\). When we do this, we get \(-2x^{2}+6x-4\).

Step 3: Perform Addition

Now, add the result of \(-2g(x)\) to the function \(f(x)\). This will lead to \((-2x^{2}+6x-4) + (2x+5)\). By simplifying this, we get \(-2x^{2}+8x+1\).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression made up of variables and coefficients, connected using addition, subtraction, and multiplication. Each term consists of a coefficient and a variable raised to a non-negative integer exponent. In the given problem, the function \( g(x) = x^2 - 3x + 2 \) is a polynomial function because it fits this definition, with a square term \( x^2 \), a linear term \( -3x \), and a constant \( 2 \).

Polynomial functions can have different degrees, which denote the highest exponent of the variable in the expression. Here, for \( g(x) \), the degree is 2, since the highest power of \( x \) is 2. Understanding polynomial functions helps with performing operations like function addition and multiplication, as seen in the original exercise.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They are key elements in mathematics used to represent real-world situations, solve equations, and perform operations on functions. The expressions are typically connected by addition, subtraction, multiplication, or division.

For example, in our exercise, the expressions \( f(x) = 2x + 5 \) and \( g(x) = x^2 - 3x + 2 \) include addition and multiplication. When dealing with such expressions, it is crucial to follow the order of operations to arrive at the correct results. Simplification often involves combining like terms or factoring.

Function Multiplication

Function multiplication involves multiplying two functions or, in some cases, multiplying each term in a function by a scalar, as seen in the original problem. The solution starts with multiplying the entire function \( g(x) = x^2 - 3x + 2 \) by \(-2\).

This operation involves distributing \(-2\) across each term within \( g(x) \), leading to:

• \(-2 \cdot x^2 = -2x^2\)
• \(-2 \cdot (-3x) = 6x\)
• \(-2 \cdot 2 = -4\)

The result is the new function \(-2x^2 + 6x - 4\). This is a foundational skill not only in algebra but also in calculus and other advanced fields of mathematics, highlighting the importance of mastering this concept.

Function Addition

Function addition is the process of adding two functions together by combining their like terms. In our example, after simplifying \(-2g(x)\), the result \(-2x^2 + 6x - 4\) is added to the function \(f(x) = 2x + 5\).

Like terms are terms that have the same variable raised to the same power. We proceed by aligning these terms:

• Combining the \(x\) terms: \(6x + 2x = 8x\)
• Adding the constants: \(-4 + 5 = 1\)
• Bring down the \(x^2\) term: \(-2x^2\)

This addition results in the simplified form \(-2x^2 + 8x + 1\). Mastery of function addition is essential for solving complex algebraic problems and understanding calculus concepts.