Problem 17 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Multiply the Functions

To find the product of two functions, multiply them together. Given functions are \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), their product is \( (f \, \cdot \, g)(x) = (2x + 1)(x - 3) \).

Step 2: Expand the Expression

Multiply the binomials from Step 1. Distribute each term in the first binomial across the second: \( 2x(x - 3) + 1(x - 3) \).

Step 3: Complete the Multiplication

Distribute and simplify: \( 2x(x) = 2x^2 \), \( 2x(-3) = -6x \), \( 1(x) = x \), \( 1(-3) = -3 \). Combining all, we get \( 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3 \).

Step 4: Determine the Domain

The domain of a function is the set of all possible input values (\( x \)) for which the function is defined. Both \( f(x) \) and \( g(x) \) are polynomials, which are defined for all real numbers. Thus, the domain of \( (f \, \cdot \, g)(x) = 2x^2 - 5x - 3 \) is all real numbers: \( (-\infty, \infty) \).

Key Concepts

Function Operations

When dealing with functions, we can perform different operations, just like with numbers. These operations include addition, subtraction, multiplication, and division of functions. Each operation combines functions in a way that allows them to interact and produce a new function. For example, if we have two functions, say \( f(x) \) and \( g(x) \), we can create a new function by:

• Adding them: \((f + g)(x) = f(x) + g(x)\)
• Subtracting them: \((f - g)(x) = f(x) - g(x)\)
• Multiplying them: \((f \cdot g)(x) = f(x) \cdot g(x)\)
• Dividing them: \((f / g)(x) = \frac{f(x)}{g(x)}\), where \(g(x) eq 0\)

Understanding these operations is crucial as they allow you to explore new relationships and behaviors of functions. In this exercise, our focus is on multiplying functions, creating a new function that encodes the interaction between \(f(x) = 2x + 1\) and \(g(x) = x - 3\). This method of multiplication effectively combines the effects of both original functions into a single expression.

Domain of a Function

The domain of a function refers to all the input values (also known as 'x' values) for which the function is defined. Understanding the domain is vital because it tells us where the function operates and produces valid outputs. For polynomial functions, like the ones in this exercise, the domain is typically all real numbers. This is because polynomial functions do not have restrictions like division by zero or negative square roots, which can limit the domain. In our function multiplication example, both \(f(x) = 2x + 1\) and \(g(x) = x - 3\) are simple linear polynomials. Each is defined for all real numbers, meaning any real number can replace \(x\) without causing an error. Therefore, when these functions are multiplied to form \((f \cdot g)(x) = 2x^2 - 5x - 3\), the domain remains all real numbers \((-, )\). It's always a good idea to check the original functions for any restrictions before performing operations. Make sure you understand why certain functions might have restricted domains to avoid errors in calculations.

Function Multiplication

Multiplying functions enhances the complexity and richness of mathematical analysis. When you multiply two functions, each term from one function is systematically distributed and multiplied by each term from the other function. This distribution results in a new polynomial. Consider \(f(x) = 2x + 1\) and \(g(x) = x - 3\) from our exercise. The multiplication process involved:

• Using the distributive property to handle the multiplication of polynomials step by step.
• First, multiplying each term of \(f(x)\) with every term in \(g(x)\).
• The method is similar to using the FOIL rule (First, Outer, Inner, Last) if both functions are binomials.

For our functions, the multiplication process is:

• First: \(2x \cdot x = 2x^2\)
• Outer: \(2x \cdot (-3) = -6x\)
• Inner: \(1 \cdot x = x\)
• Last: \(1 \cdot (-3) = -3\)

Combining these products results in a new function: \(2x^2 - 5x - 3\). This systematic multiplication is essential for understanding how combined functional behaviors translate into a single mathematical expression. It provides insights into how different elements in functions interact numerically and graphically.