Problem 76 - Elementary and Intermediate Algebra

Question

Let \(f(x)=x^{2}-1\) and \(g(x)=x^{2}-4 .\) Find each function and give its domain. $$ g \cdot f $$

Step-by-Step Solution

Answer

The product function is \((g \cdot f)(x) = x^4 - 5x^2 + 4\) with domain \((-\infty, +\infty)\).

Step 1: Identify the Functions

The given functions are \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \). We need to find the product of these two functions.

Step 2: Multiply the Functions

The product of the functions is given by \[(g imes f)(x) = g(x) \cdot f(x) = (x^2 - 4)(x^2 - 1)\].Expand the product:\[(x^2 - 4)(x^2 - 1) = x^2(x^2 - 1) - 4(x^2 - 1)\].Calculate each part:\[= x^4 - x^2 - 4x^2 + 4\], which simplifies to \[= x^4 - 5x^2 + 4\].Therefore, the product function \((g imes f)(x) = x^4 - 5x^2 + 4\).

Step 3: Determine the Domain

The domain of a polynomial function is all real numbers because polynomials are defined for every real number. Both \(f(x)\) and \(g(x)\) are polynomials. Hence, the domain of \((g \cdot f)(x) = x^4 - 5x^2 + 4\) is all real numbers, denoted by \((-\infty, +\infty)\).

Key Concepts

Function Multiplication

Function multiplication involves multiplying two functions together to form a new function. When we have functions like \( f(x) \) and \( g(x) \), the product function is represented as \((g \cdot f)(x)\), meaning \( g(x) \cdot f(x) \). By performing this operation, we are essentially combining the terms of the functions through multiplication.

In our exercise, we multiplied \( f(x) = x^2 - 1 \) by \( g(x) = x^2 - 4 \). Let's break it down step by step:

• First, multiply \( x^2 \) from \( f(x) \) with each term of \( g(x) \), producing \( x^4 - x^2 \).
• Next, multiply \(-4\) with each term of \( g(x) \), producing \(-4x^2 + 4\).
• Combine these results: \(x^4 - x^2 - 4x^2 + 4\); which simplifies to \(x^4 - 5x^2 + 4\).

Each term is handled carefully, ensuring no step is skipped. This careful expansion ensures accuracy in creating a new, single polynomial from the two original expressions.

Domain of Functions

The domain of a function refers to all possible input values (or \( x \)-values) for which the function is defined. In simpler terms, it tells us what kind of numbers can "go into" our function without causing problems like division by zero or the square root of a negative.

Polynomial functions, like \( f(x) = x^2 - 1 \) and \( g(x) = x^2 - 4 \), have very straightforward domains. One of the great things about polynomial functions is that they are defined for all real numbers. This is because there are no restrictions such as divisions or negative radicands in square roots. Therefore, the domain of any polynomial function in this exercise, including the product \((g \cdot f)(x) = x^4 - 5x^2 + 4\), extends to all real numbers.

We express this as \((-\infty, +\infty)\), meaning any real number can be substituted into the function.

Real Numbers

Real numbers encompass all the numbers that you can think of, except imaginary numbers. They include natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

• Natural Numbers: These are positive integers starting from 1, 2, 3, and so on.
• Whole Numbers: Similar to natural numbers, but they include 0.
• Integers: These include all whole numbers and their negative counterparts.
• Rational Numbers: These are numbers that can be expressed as a quotient of two integers (e.g., 1/2, 4/3).
• Irrational Numbers: These cannot be expressed as a simple fraction, such as \( \sqrt{2} \) or \( \pi \).

In the context of polynomial functions, understanding real numbers helps us see the entirety of possible inputs that define the domain. Since polynomial functions, including the product \((g \cdot f)(x)\), can take any real number as input, knowing the concept of real numbers aids in grasping why the domain is \((-\infty, +\infty)\). Every real number is applicable here, with no exceptions!