Problem 34 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$f(x)=3 x-2$$

Step-by-Step Solution

Answer

The domain and range of the function \( f(x) = 3x - 2 \) are both all real numbers.

Step 1: Identify the Function

The function given is \( f(x) = 3x - 2 \), which is a linear function.

Step 2: Analyze the Domain

For this function, \( x \) can be any real number, so the domain is all real numbers.

Step 3: Analyze the Range

For a linear equation, the range, like the domain, is also all real numbers as there is no restriction on the output of this function.

Key Concepts

Linear Functions

Linear functions are a foundational concept in algebra and graphing. They have the form of \( y = mx + b \), where \( m \) and \( b \) represent constants. The variable \( m \) is known as the slope, and it indicates how steep the line is. Meanwhile, \( b \) is the y-intercept, the point at which the line crosses the y-axis.

In the case of the function \( f(x) = 3x - 2 \), we can see that it is indeed a linear function. The slope here is 3, meaning that for every unit increase in \( x \), the value of \( f(x) \) rises by 3 units. The y-intercept is -2, indicating that the line will cross the y-axis at the point (0, -2).

Understanding the characteristics of linear functions is crucial because it helps in predicting behavior of change. Whether you're analyzing trends or calculating rates, knowing how to work with linear functions will provide clear insights into relationships between variables.

Real Numbers

The concept of real numbers comprises all the numbers that can be found on the number line. This includes both rational numbers (like fractions and integers) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)). In our everyday lives, when we measure, count, or observe continuous quantities, we're constantly interacting with real numbers.

When analyzing a function, particularly the domain, we specify what values \( x \) can take. In the case of linear functions, like the one given \( f(x) = 3x - 2 \), the domain is all real numbers because there is no value of \( x \) for which this function becomes undefined. This comprehensive domain is essential to many mathematical models that assume any input value is possible, which is often the case in real-world scenarios.

Function Analysis

Function analysis is essentially about understanding the behavior of functions: how they change, what values they produce, and the restrictions on those values. This kind of analysis typically involves determining the domain and range of functions.

The domain is the set of all possible input values (\( x \)-values) for the function, while the range is the set of all possible outputs (\( y \)-values). For the linear function in our example (\( f(x) = 3x - 2 \)), both the domain and range are all real numbers. This means that you can plug any real number into \( f(x) \) and get a real number out.

Even when a function seems simple, like a straight line on a graph, performing a thorough analysis can reveal important information about its characteristics, such as how it behaves over different intervals and its interaction with other mathematical entities in a given system.