Problem 14 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

(f-g)(x) = x + 4 with domain \(\mathbb{R}\).

Step 1: Understand the Functions

We are given two functions: \(f(x) = 2x + 1\) and \(g(x) = x - 3\). We need to find the result of \(f - g\), which means we subtract the function \(g(x)\) from the function \(f(x)\). Subtraction of functions involves subtracting their expressions.

Step 2: Subtract the Functions

To subtract \(g(x)\) from \(f(x)\), use the equation \((f-g)(x) = f(x) - g(x)\). Substitute the expressions for \(f(x)\) and \(g(x)\):\[(f-g)(x) = (2x + 1) - (x - 3)\]

Step 3: Simplify the Expression

Simplify the expression obtained in Step 2:\[(f-g)(x) = 2x + 1 - x + 3\]Combine like terms:\[(f-g)(x) = x + 4\]

Step 4: Determine the Domain

Both \(f(x) = 2x + 1\) and \(g(x) = x - 3\) are linear functions, and their domain is all real numbers, \(\mathbb{R}\). Since \(f-g\) is also a linear function, its domain is the same, \(\mathbb{R}\).

Key Concepts

Understanding Linear Functions

Linear functions are among the simplest and most commonly used types of functions in mathematics. They can be written in the standard form of \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. These functions represent straight lines when plotted on a graph.

For example, in the function \(f(x) = 2x + 1\), the slope is 2, indicating that for each unit increase in \(x\), the value of the function increases by 2. The y-intercept is 1, meaning the line crosses the y-axis at (0,1). Likewise, in \(g(x) = x - 3\), the slope is 1, and the y-intercept is -3.

Linear functions are fundamental because they form the basis for understanding more complex functions. They are used in a variety of fields from physics to economics, representing constant rates of change.

• They grow or decrease consistently.
• Are defined by two main parameters: slope and y-intercept.
• Model real-world scenarios with steady changes.

Exploring the Domain of a Function

The domain of a function is the set of all possible input values (usually \(x\) values) for which the function is defined. For linear functions like \(f(x) = 2x + 1\) and \(g(x) = x - 3\), the domain is typically all real numbers, denoted \(\mathbb{R}\).

This means that for any value of \(x\), we can substitute it into the function, and we will always get a valid output \(f(x)\) or \(g(x)\). There are no restrictions, such as division by zero, which simplify things significantly when working with linear functions.

In practical terms, if you are asked to find the domain of a linear function, you can confidently state that it is all real numbers. Keep in mind that while linear functions naturally have infinite domains, other types of functions may have restricted domains.

• The domain ensures functions are well-defined.
• Linear functions' domains are always \(\mathbb{R}\).
• Domains become critical in functions with restrictions, such as square roots or fractions.

Performing Subtraction of Functions

Subtraction of functions involves finding the difference between two given functions over their domain. For functions such as \(f(x) = 2x + 1\) and \(g(x) = x - 3\), the subtraction \(f - g\) is carried out by subtracting the expressions of the functions. This results in a new function:

\[(f-g)(x) = f(x) - g(x) = (2x + 1) - (x - 3) = x + 4\]

To perform the subtraction, combine like terms from both functions, which will simplify the expression further. Here, we subtract like terms: substract \(x\) from \(2x\), and add constants \(1\) and \(3\) together. This subtraction operation results in the function \((f-g)(x) = x + 4\), another linear function.

Subtraction of functions, like addition, doesn't alter the domain, as it remains the set of all real numbers if original functions were well-defined everywhere.

• Write the expression for \((f-g)(x)\).
• Subtract corresponding terms.
• Simplify to find the resulting function.