Problem 20 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The function \( g \cdot f(x) = 2x^2 - 5x - 3 \) has the domain \((-\infty, \infty)\).

Step 1: Understanding the Functions

We're given two functions, \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \). The task is to find the new function formed by their multiplication, \( g \cdot f \), and to determine its domain.

Step 2: Performing the Multiplication

To find \( g \cdot f \), multiply the two functions: \((g \cdot f)(x) = g(x) \cdot f(x) = (x - 3)(2x + 1) \). Distribute the terms: \( x(2x + 1) - 3(2x + 1) = 2x^2 + x - 6x - 3 = 2x^2 - 5x - 3 \). Thus, the new function is \( g \cdot f(x) = 2x^2 - 5x - 3 \).

Step 3: Determining the Domain

The domain of a function \( h(x) \) is the set of all real numbers \( x \) for which \( h(x) \) is defined. Since \( g \cdot f(x) \) is a polynomial, it is defined for all real numbers. Thus, the domain of \( g \cdot f \) is all real numbers, or \( (-\infty, \infty) \).

Key Concepts

Function Multiplication

Function multiplication is an operation that combines two functions to form a new function. This is represented as \( (g \cdot f)(x) \), where each function is evaluated at \( x \) and then multiplied together. When you have two functions, say \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), you create a new function by multiplying these two expressions together. This involves distributing each term in the first function across each term in the second function.Here's a simple breakdown of the process:

• Start with \( (g \cdot f)(x) = g(x) \cdot f(x) = (x - 3)(2x + 1) \).
• Apply the distributive property: multiply \( x \) by each term in \( 2x + 1 \), and \( -3 \) by each term in \( 2x + 1 \).
• Combine like terms, giving you the expanded form: \( 2x^2 - 5x - 3 \).

This resulting polynomial \( 2x^2 - 5x - 3 \) becomes the new function for \( g \cdot f(x) \). Function multiplication like this is a fundamental concept in algebra, useful for combining different functions into a single equation.

Domain of a Function

The domain of a function refers to all the possible input values (or \( x \)-values) for which the function is defined. For polynomial functions, like \( g \cdot f(x) = 2x^2 - 5x - 3 \), the domain includes all real numbers.Here’s why the domain is crucial:

• Understanding the domain helps us know the values for which the function formula can be applied without any mathematical error, such as division by zero or taking the square root of negative numbers.
• For polynomial functions, the terms don’t include any division by variable expressions or other operations that might limit inputs.

Therefore, the domain for \( g \cdot f(x) \) is all real numbers, written as \((-\infty, \infty)\). This means you can plug in any real number into the function, and it will give you a valid output without any restrictions.

Real Numbers

Real numbers form the set of numbers that include all the rationals (fractions) and irrationals (like \( \pi \) and \( \sqrt{2} \)). They are denoted as \( (-\infty, \infty) \), representing every point on the number line. Understanding real numbers is essential when discussing the domain of functions, especially polynomial functions, which are defined for all real numbers. When we talk about real numbers, we refer to:

• Whole numbers like 1, 2, 3, etc., including negatives and zero.
• Fractions and decimals, such as 1/2 or 0.75.
• Irrational numbers, which can't be expressed as exact fractions, like \( \sqrt{2} \) or \( \pi \).

In the context of polynomial functions, since they can take any real number and provide a valid output, their domain is the entire set of real numbers. This broad applicability makes them very versatile in mathematical modeling and problem-solving across various fields.