Problem 68 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=2 x^{2}+1 .\) Find each function and give its domain. $$ f \cdot g $$

Step-by-Step Solution

Answer

The product \( (f \cdot g)(x) = 6x^3 - 4x^2 + 3x - 2 \) and its domain is all real numbers.

Step 1: Understanding the Functions

We are given two functions: \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \). Our task is to find the product of these two functions, \( (f \cdot g)(x) \).

Step 2: Multiply the Functions

To find \( (f \cdot g)(x) \), we need to multiply \( f(x) \) and \( g(x) \). This results in:\[(f \cdot g)(x) = f(x) \cdot g(x) = (3x - 2)(2x^2 + 1)\]

Step 3: Apply the Distributive Property

Using the distributive property, we expand the expression:\[(f \cdot g)(x) = 3x(2x^2) + 3x(1) - 2(2x^2) - 2(1)\]This simplifies to:\[(f \cdot g)(x) = 6x^3 + 3x - 4x^2 - 2\]

Step 4: Simplify the Expression

Rearrange the expanded expression by combining like terms:\[(f \cdot g)(x) = 6x^3 - 4x^2 + 3x - 2\]

Step 5: Determine the Domain

The domains of \( f(x) \) and \( g(x) \) are all real numbers since there are no restrictions like division by zero or square roots of negative numbers. Hence, the domain of \( (f \cdot g)(x) \) is also all real numbers, \( (-\infty, \infty) \).

Key Concepts

Function Multiplication

Multiplication of functions is an important operation in algebra, allowing us to combine two functions into one new function. Given two functions, say \( f(x) \) and \( g(x) \), their product is represented as \((f \cdot g)(x)\), which is simply the product of their outputs for the same input \( x \).
To multiply functions, you follow these steps:

• Substitute each function into the multiplication equation, \( f(x) \cdot g(x) \).
• Use basic algebraic properties, such as the distributive property, to simplify the expression fully.

The goal is to transform the expression into a simplified polynomial by combining and arranging terms as necessary.
In the example given, where \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \), multiplying them required distributing each term in \( f(x) \) by each term in \( g(x) \), ultimately simplifying to a new function: \( 6x^3 - 4x^2 + 3x - 2 \). Remember, careful expansion and simplification are key to accurate function multiplication.

Polynomial Functions

Polynomial functions form a fundamental part of algebra. They consist of one or more terms, each including a variable raised to a non-negative integer exponent and a coefficient. A typical polynomial looks like \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients and \( n \) indicates the degree, or highest power, of the polynomial.
Key features of polynomials include:

• The highest degree of a term determines the degree of the polynomial.
• Coefficients can be any real numbers.
• Polynomials are continuously differentiable and smooth functions; they don't have breaks, holes, or asymptotes, unlike some other function types.

In our example, after multiplying \( f(x) \) and \( g(x) \), we ended up with a cubic polynomial: \( 6x^3 - 4x^2 + 3x - 2 \). This polynomial has a degree of 3, as the highest power of \( x \) is 3. Each term represents a unique combination of coefficients and powers of \( x \).

Domain of a Function

The domain of a function refers to the complete set of possible values of the independent variable, \( x \), for which the function is defined. Understanding the domain is crucial because it tells us where the function can operate without running into mathematical issues like division by zero or taking a square root of a negative number.
When considering polynomials:

• Polynomials inherently have a domain that includes all real numbers because their operations—addition, subtraction, multiplication—don't impose restrictions.
• The domain of a polynomial is typically expressed as \( (-\infty, \infty) \).

In our example, the polynomial function \( 6x^3 - 4x^2 + 3x - 2 \) was derived by multiplying two functions, \( f(x) \) and \( g(x) \). Since both are polynomials, their product is also a polynomial with an unlimited domain, meaning it exists at every point along the real number line. Therefore, the domain for \((f \cdot g)(x)\) covers all real numbers.