Problem 53 - Algebra 2

Question

Let \(f(x)=3 x^{2}+2 x-8\) and \(g(x)=x+2 .\) Perform each function operation and then find the domain. $$ f(x) \cdot g(x) $$

Step-by-Step Solution

Answer

The function \(f(x) \cdot g(x) = 3x^{3} + 8x^{2} - 4x -16\) and the domain of this function is all real numbers, \(R\).

Step 1: Perform the Operation

First, perform the operation \(f(x) \cdot g(x)\). Multiply the functions \(f(x)\) and \(g(x)\) i.e. \((3 x^{2}+2 x-8) \cdot (x+2)\). Try to distribute each element of the polynomial of the function \(f(x)\) with each term in the function \(g(x)\). You should get \(3x^{3} + 6x^{2} + 2x^{2} + 4x - 8x -16\). Combine the like terms, and you will get \(3x^{3} + 8x^{2} - 4x -16\).

Step 2: Determine the Domain

Now, since the function obtained from the above step is a polynomial function, it is defined for all real values. Therefore the domain is all real numbers, \(R\).

Step 3: Final Answer

From the above steps, the function \(f(x) \cdot g(x)\) equals to \(3x^{3} + 8x^{2} - 4x -16\). And the domain is all real numbers, \(R\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients. These expressions involve operations of addition and multiplication, combining variables raised to whole number powers. For instance, a polynomial function can take the form of \( f(x) = 3x^2 + 2x - 8 \).

• Key features include their straightforward behavior; they are continuous and smooth.
• Common operations include addition, subtraction, multiplication, and sometimes division.
• Each term in a polynomial function has a degree, indicated by the exponent of the variable.

Polynomials are simple yet remarkably powerful, used across various mathematical applications like modeling curves or solving equations. Understanding them helps one grasp more complicated mathematical concepts like calculus.

Domain of a Function

The domain of a function refers to the set of all possible inputs (or \( x \)-values) that a function can accept without leading to any undefined conditions. For polynomial functions, the domain is often all real numbers.

• This is because you can plug any real number into the function without causing issues like division by zero or taking the square root of a negative number, which are common domain restrictions in other types of functions.
• In mathematical notation, we express this as the domain being equal to \( \mathbb{R} \), representing all real numbers.
• It is crucial to identify the domain to avoid undefined mathematical operations during function solving and graphing processes.

Always determine the domain first to ensure you correctly address the entire range where functions behave predictably.

Multiplication of Functions

Multiplicating functions is a fundamental operation in algebra, useful for combining multiple function outputs. When multiplying two functions like \( f(x) \times g(x) \), you multiply each term of one function by every term of the other.

For example, given \( f(x) = 3x^2 + 2x - 8 \) and \( g(x) = x + 2 \), to find \( f(x) \cdot g(x) \):

• Distribute each term of \( f(x) \) across \( g(x) \): \( (3x^2)(x + 2) + (2x)(x + 2) + (-8)(x + 2) \).
• This results in \( 3x^3 + 6x^2 + 2x^2 + 4x - 8x - 16 \).
• Simplify by combining like terms: \( 3x^3 + 8x^2 - 4x - 16 \).

Through multiplication, you create a new polynomial that integrates the behavior of both original functions. This method amplifies the utility of each single function into a broader application.

Real Numbers

Real numbers encompass all numbers that can be found on the number line, including both rational numbers (such as 1/2 or 3) and irrational numbers (like \( \pi \) or \( \sqrt{2} \)).

• They include integers, fractions, and decimals, covering both positive and negative numbers, as well as zero.
• Real numbers form the most extensive category of numbers used in everyday mathematics, except for imaginary and complex numbers.
• In context of polynomial functions and domains, real numbers are critical because polynomial functions are typically defined across all real numbers.

Recognizing the role of real numbers solidifies the understanding of domains for polynomial and other types of functions. This comprehensive set of numbers ensures versatile applicability in mathematical operations and real-world problem-solving.