Problem 14 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Function Definitions

We begin by examining the functions given: \(f(x) = 2x + 1\) and \(g(x) = x - 3\). These are both linear functions.

Step 2: Calculate the Difference of Functions

The problem requires finding \(f-g\), which means we need to subtract the function \(g(x)\) from \(f(x)\). So, \((f-g)(x) = f(x) - g(x) = (2x + 1) - (x - 3)\).

Step 3: Simplify the Expression

Distribute and combine like terms in the expression: \((2x + 1) - (x - 3) = 2x + 1 - x + 3 = (2x - x) + (1 + 3)\). This simplifies to \(x + 4\).

Step 4: Determine the Domain

Both \(f(x)\) and \(g(x)\) are linear functions, which are defined for all real numbers. Thus, the domain of \(f-g\) is also all real numbers, \(\mathbb{R}\).

Key Concepts

Linear Functions

Linear functions are one of the most straightforward types of functions in algebra, yet they are incredibly powerful. A linear function takes the form \(f(x) = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept. This means that the graph of a linear function is always a straight line, and the slope \(m\) indicates how steep this line is. Atoms of linear functions include:

• Slope \(m\): This dictates the direction and steepness of the line. A positive slope means the line rises from left to right, and a negative slope means it falls.
• Y-intercept \(b\): This is the point where the line crosses the y-axis. It gives us a starting value where \(x=0\).

In our example, with \(f(x) = 2x + 1\), the slope is \(2\), signifying that for every unit increase in \(x\), \(f(x)\) increases by 2 units. The y-intercept is \(1\), meaning the line crosses the y-axis at \(y=1\). Understanding these two parameters allows anyone to visualize the line's behavior easily.

Function Operations

Operations on functions are similar to basic arithmetic operations that you might be familiar with, such as addition, subtraction, multiplication, and division. These operations allow us to create new functions by combining existing ones.In the given exercise, we are subtracting one function from another, which is a common operation called the difference of functions. To find \((f-g)(x)\), you subtract the second function \(g(x)\) from the first function \(f(x)\):\[(f-g)(x) = f(x) - g(x)\]Substituting the expressions from our functions, we get:\[(f-g)(x) = (2x + 1) - (x - 3)\]It is crucial to distribute the subtraction across \(g(x)\) properly:

• The expression \((x - 3)\) becomes \(-x + 3\).
• Combining like terms \((2x - x) \) and adding the constants \((1 + 3)\) gives \((f-g)(x) = x + 4\).

Understanding function operations like these is key to manipulating and analyzing algebraic functions.

Domain of a Function

The domain of a function refers to all possible input values \(x\) for which the function \(f(x)\) is defined. For most purposes with linear functions, this set is all real numbers, represented by \(\mathbb{R}\). Linear functions such as \(f(x) = 2x + 1\) or \(g(x) = x - 3\) have no restrictions, meaning there aren't any values of \(x\) that you can't substitute into the function. This is because linear equations don't have any variables in the denominator (which could make them undefined) nor do they include square roots or logarithms (which could impose additional restrictions).In the solution, both \(f(x)\) and \(g(x)\) are identified as being defined for all real numbers, thus:

• The domain of \(f(x) = 2x + 1\) is \(\mathbb{R}\).
• The domain of \(g(x) = x - 3\) is also \(\mathbb{R}\).
• Consequently, the domain of their difference \((f-g)(x) = x + 4\) remains \(\mathbb{R}\).

Grasping the concept of function domains ensures that we know the valid inputs, preventing errors in calculation or analysis.