Problem 68 - 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 function is \( f \cdot g = 6x^3 - 4x^2 + 3x - 2 \) with domain \((-\infty, \infty)\).

Step 1: Define the Functions

We are given the functions \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \). In this step, we simply note down these given functions, which we will use in our calculations.

Step 2: Calculate the Product of the Functions

To find \( f \cdot g \), we need to multiply the two functions. This means we will calculate \((3x - 2)(2x^2 + 1)\). We will do this using the distributive property: \[ f \cdot g = (3x - 2)(2x^2 + 1) = 3x(2x^2 + 1) - 2(2x^2 + 1) \] Now expand each term: \[ = 3x \cdot 2x^2 + 3x \cdot 1 - 2 \cdot 2x^2 - 2 \cdot 1 \] \[ = 6x^3 + 3x - 4x^2 - 2 \] Thus, \( f \cdot g = 6x^3 - 4x^2 + 3x - 2 \).

Step 3: Determine the Domain

The domain of a function is the set of all possible input values \( x \) that do not cause any undefined behavior in the function. Both \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \) are polynomials. The domain of any polynomial function is all real numbers, i.e., \( (-\infty, \infty) \). Thus, the domain of \( f \cdot g \) is also \( (-\infty, \infty) \).

Key Concepts

polynomial functions

A polynomial function is one of the most fundamental types of functions you'll encounter in algebra. They are expressed as sums of terms consisting of variables raised to whole number exponents. Typically, a polynomial is of the form:

• Constant: such as 5 or -3
• Linear term: for example, 2x
• Quadratic term: like 4x^2
• Cubic term: such as -6x^3

The beauty of polynomial functions lies in their simplicity and predictability. Each term is a product of a coefficient and a variable raised to an exponent. For example, in the function \(f(x) = 3x - 2\), we have a linear polynomial, where the term 3x is the linear part and -2 is the constant term. Polynomial functions can be graphed as smooth, continuous lines/curves, reflecting their straightforward nature.
They are essential for modeling many real-world situations, from simple to complex ones. Understanding polynomials is crucial as they form the building blocks for more advanced mathematical concepts.

function multiplication

Function multiplication involves taking two functions and combining them by multiplying their outputs for each input value. This operation is not merely multiplying numbers; it means multiplying the expressions and simplifying them.
For instance, if you have functions \(f(x) = 3x - 2\) and \(g(x) = 2x^2 + 1\), to find \(f \cdot g\), you multiply these expressions. You apply the distributive property:

• Multiply each term in the first function by every term in the second function
• Simplify the resulting expression

In this example, \((3x - 2)(2x^2 + 1)\) expands and simplifies to \(6x^3 - 4x^2 + 3x - 2\). This becomes a new polynomial function that represents the product of \(f\) and \(g\). This procedure allows us to better understand the behavior of combined functions, offering insights into system interactions or more complex models in applied settings.

function domain

The domain of a function is all the input values \(x\) for which the function is defined and provides an output. For polynomial functions like \(f(x) = 3x - 2\) and \(g(x) = 2x^2 + 1\), these functions are defined for all real numbers.
Polynomials are

• Continuous: They don't have breaks or holes in their graphs
• Real-number inclusive: They can accept any real number as an input
• Well-defined everywhere: There's no limit to the inputs concerning undefined operations like division by zero

Thus, their domain is \((-\infty, \infty)\). This means you can substitute any real number into these functions without encountering issues. When working with combinations of polynomials, such as in \(f \cdot g\), the domain remains the real numbers. Understanding domains helps ensure that all potential inputs are considered when analyzing function behavior.