Problem 17 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\((f \cdot g)(x) = 2x^2 - 5x - 3\); domain: all real numbers.

Step 1: Understand Function Notation

The goal is to calculate the product of two functions. Given that \(f(x) = 2x + 1\) and \(g(x) = x - 3\), the function \((f \cdot g)(x)\) means \(f(x) \times g(x)\).

Step 2: Multiply the Functions

Calculate \((f \cdot g)(x)\) by multiplying \(f(x)\) and \(g(x)\):\((f \cdot g)(x) = (2x + 1)(x - 3)\).Now, expand this expression.

Step 3: Expand the Expression

Use the distributive property to expand:\((2x + 1)(x - 3) = 2x(x) + 2x(-3) + 1(x) + 1(-3)\).Simplify each term: \(2x^2 - 6x + x - 3\).

Step 4: Simplify the Expression

Combine like terms:\(2x^2 - 6x + x - 3 = 2x^2 - 5x - 3\).This is the expression for \((f \cdot g)(x)\).

Step 5: Determine the Domain of the Product Function

The domain of a function is the set of all possible input values (\(x\)) that make the function defined. Both \(f(x) = 2x + 1\) and \(g(x) = x - 3\) are linear functions, so they are defined for all real numbers. Thus, the domain of their product \((f \cdot g)(x) = 2x^2 - 5x - 3\) is also all real numbers, \(( -\infty, +\infty)\).

Key Concepts

Function Notation

Understanding function notation is crucial when dealing with function operations like multiplication. Function notation gives clarity on how functions are expressed and evaluated. For a function, say \( f \), we write \( f(x) \) to mean the function \( f \) evaluated at \( x \). This replaces the variable in the function's expression with \( x \) to specify input values. In the exercise, \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \) are given, using function notation to emphasize what happens when particular values are substituted for \( x \). This is particularly useful when you're asked to calculate expressions like \((f \cdot g)(x)\), which involves substituting \( x \) into another expression formed by multiplying \( f(x) \) and \( g(x) \). As such, function notation helps to communicate the process of evaluating and manipulating functions in a concise way.

Distributive Property

In mathematics, the distributive property is a helpful tool for multiplying expressions, especially when dealing with polynomials or, in our case, when multiplying functions. It states that for any three numbers or expressions \( a, b, \) and \( c \), the expression \( a(b + c) \) is equal to \( ab + ac \). This property was applied during the expansion of \((f \cdot g)(x) = (2x + 1)(x - 3)\). To expand this, you multiply each term in the first polynomial by each term in the second.

Let's break it down step by step:

• Multiply the first terms: \(2x \cdot x = 2x^2\)
• Multiply the outer terms: \(2x \cdot (-3) = -6x\)
• Multiply the inner terms: \(1 \cdot x = x\)
• Multiply the last terms: \(1 \cdot (-3) = -3\)

This process is often summarized using the FOIL method (First, Outer, Inner, Last). Finally, you combine similar terms, in this case, resulting in \(2x^2 - 5x - 3\). By mastering the distributive property, you gain a powerful skill in algebra and beyond.

Domain of a Function

The domain of a function is a set of all possible input values (often \( x \)), for which the function is defined. Determining the domain is crucial because it tells you where the function can be applied without causing undefined operations. In general, linear functions like \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \) have a domain of all real numbers, \((-\infty, \infty)\).

When these functions are multiplied to form \((f \cdot g)(x) = 2x^2 - 5x - 3\), the resulting function is a quadratic polynomial, which similarly is defined for all real numbers. This means there are no restrictions on \( x \) for \( (f \cdot g)(x) \). Polynomials of any degree, unless specified by additional conditions (like division by zero or square roots of negative numbers), usually cover the entire number line. Always check the form of the function to define its domain correctly.