Problem 16 - Algebra 2

Question

Let \(f(x)=2 x^{2}+x-3\) and \(g(x)=x-1 .\) Perform each function operation and then find the domain. $$ f(x) \cdot g(x) $$

Step-by-Step Solution

Answer

The result of the function operation \(f(x) \cdot g(x)\) is \(h(x) = 2x^{3} - x^{2} - 4x + 3\) and its domain is all real numbers \(\mathbb{R}\).

Step 1: Perform Function Operation

The function operation is multiplication. Hence, we will multiply the functions \(f(x)\) and \(g(x)\). Let's call the resulting function \(h(x)\). Hence, \(h(x) = f(x) \cdot g(x)\). So \(h(x) = (2x^{2} + x - 3) \cdot (x - 1) = 2x^{3} - 2x^{2} + x^{2} - x - 3x + 3 = 2x^{3} - x^{2} - 4x + 3\

Step 2: Find the Domain

The domain of a function is the set of all real numbers for which the function is defined. A cubic function like \(h(x)\) is defined for all real numbers. Therefore, the domain of \(h(x)\) is all real numbers, which is usually denoted as \(\mathbb{R}\).

Key Concepts

Function Operations

In mathematics, function operations refer to various ways of combining functions. These operations include addition, subtraction, multiplication, and division. Each operation results in a new function that is derived from the initial functions.

When you perform an operation such as multiplication on functions, you combine each term of the first function with each term of the second function. Consider two functions, say \(f(x)\) and \(g(x)\). If you want to find the product \(f(x) \cdot g(x)\), you will multiply each term of \(f(x)\) with each term of \(g(x)\). This process is similar to how you perform the distributive property with numbers.

For example, if \(f(x) = 2x^2 + x - 3\) and \(g(x) = x-1\), you multiply each term of \(f(x)\) by each term of \(g(x)\) and then combine like terms. This results in a new function \(h(x) = 2x^3 - x^2 - 4x + 3\).

This multistep process helps to form a composite function, which expresses a new relationship between the input variable and output through the combined effects of the original functions.

Domain of a Function

The domain of a function is a crucial aspect of understanding how functions behave. When we talk about the domain, we are referring to all the possible input values \(x\) for which the function \(f(x)\) is defined.

Most polynomial functions, like quadratic, cubic, and higher-order polynomials, are defined for all real numbers, meaning their domain is \(\mathbb{R}\). This is because these functions don't involve operations that can restrict their input, such as division by zero or taking the square root of a negative number.

For the function \(h(x) = 2x^3 - x^2 - 4x + 3\), a cubic polynomial, there are no restrictions preventing any real number from being an input. Since cubic functions are smooth and continuous across all real numbers, the domain is all real numbers \(\mathbb{R}\).

Understanding the domain of a function helps students anticipate what values can be used as inputs, leading to clearer problem-solving and function manipulation.

Cubic Functions

Cubic functions are polynomial functions of degree three, generally expressed in the form \(f(x) = ax^3 + bx^2 + cx + d\). These functions can take on a variety of shapes depending on their coefficients and can display interesting characteristics such as their curve and inflection points.

One of the hallmarks of cubic functions is their ability to model changes that are not linear or quadratic. They can represent more complex relationships characterized by how they smoothly curve upwards and downwards, potentially passing through real roots (or zeros) up to three times.

Cubic functions are defined for all real numbers, meaning their domain is \(\mathbb{R}\). This property makes them versatile for various applications in physics, engineering, and economics, among other fields.

Understanding cubic functions includes identifying their turning points, which are places where the function switches from increasing to decreasing or vice versa. Learning about these features can provide critical insights into their graphical behavior and wider applications.