Problem 23 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x\) and \(g(x)=4 x .\) Find each function and give its domain. $$ g-f $$

Step-by-Step Solution

Answer

The function is \(x\) and its domain is all real numbers \((-\infty, \infty)\).

Step 1: Understand Function Notation

The functions given are \(f(x) = 3x\) and \(g(x) = 4x\). We want to find the function \(g(x) - f(x)\). Understanding this means we need to subtract the expression for \(f(x)\) from the expression for \(g(x)\).

Step 2: Subtract the Functions

To find \(g(x) - f(x)\), we subtract the two functions:\[g(x) - f(x) = (4x) - (3x)\]Simplifying this expression, we get:\[g(x) - f(x) = 4x - 3x = x\]

Step 3: Determine the Domain

The domain of a function is the set of all possible input values \(x\) for which the function is defined. Both \(f(x) = 3x\) and \(g(x) = 4x\) are linear functions, which are defined for all real numbers. Therefore, \(g(x) - f(x) = x\) also has a domain of all real numbers. Thus, the domain is \((-\infty, \infty)\).

Key Concepts

Linear Functions

Linear functions are a fundamental concept in mathematics, especially in algebra, where they appear frequently. These functions are characterized by their simple structure and straight-line graphs. A linear function is generally expressed in the form of \( f(x) = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept of the function. This straightforward equation describes how the function changes: the slope \( m \) indicates the rate of change or steepness of the line, while the intercept \( b \) tells us where the line crosses the y-axis.
Linear functions are called "linear" because their graphs are straight lines. The slope \( m \) is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Hence, if you increase \( x \) by one unit, the function value increases by \( m \) units.
The simplicity of linear functions often makes them a starting point for understanding more complex mathematical concepts, such as higher-degree polynomials and calculus. Identifying linear relationships between variables is crucial in fields ranging from economics to engineering.

Domain of a Function

Understanding the domain of a function is crucial for comprehending what values are valid input for the function. The domain essentially answers the question: "What x-values can I put into this function and still get a sensible output?" In most cases, determining the domain depends on the type of function you are dealing with.
For linear functions, such as \( f(x) = 3x \) and \( g(x) = 4x \), the domain includes all real numbers. This is because linear functions can accept any real number as input and will always produce a defined output. With no restrictions or exceptions (like division by zero or the square root of a negative number), linear functions are straightforward with their domains.
More complex functions might have limited domains. For instance, fractions with variables in the denominator can’t have values that make the denominator zero. Likewise, square roots of negative values aren't defined within the real number system, which affects their domains. Always consider the definition and structure of a function to identify its domain effectively.

Function Notation

Function notation is a way to express mathematical functions in a clear and concise way, which is especially helpful when dealing with multiple functions. In essence, it provides a name to a relation and specifies how each input relates to an output. This is typically shown as \( f(x) \), where "\( f \)" is the name of the function and "\( x \)" is the variable or input.
By using function notation, you easily differentiate between various functions. For example, \( f(x) = 3x \) and \( g(x) = 4x \) are distinct because they are named differently, making it simple to identify which expression belongs to which function.
Function notation also provides a convenient way to perform operations on functions, like addition, subtraction, multiplication, or division. When you are asked to find \( g(x) - f(x) \), you are being prompted to subtract the expressions of these functions. This is a clean and efficient method to handle functions in algebra.
Overall, function notation's primary advantage is clarity, allowing mathematicians to work systematically with complex problems.