Problem 43 - Algebra and Trigonometry

Question

Find the domain of the function. $$ f(x)=2 x $$

Step-by-Step Solution

Answer

The domain of \( f(x) = 2x \) is all real numbers: \((-\infty, \infty)\).

Step 1: Understanding the Function Type

The function given is a linear function: \( f(x) = 2x \). Linear functions are defined for all real numbers unless there is a specific restriction given, such as a fraction with a zero in the denominator or a square root of a negative number.

Step 2: Identifying Domain Limitations

For the function \( f(x) = 2x \), there are no denominators and no square roots. Therefore, none of these common restrictions apply. There are no points where the function is undefined.

Step 3: Determining the Domain

Since there are no restrictions on \( x \), the domain of the linear function \( f(x) = 2x \) is all real numbers. It can be expressed in interval notation as \((-\infty, \infty)\).

Key Concepts

Linear Function

A linear function is one of the simplest forms of a mathematical function. It is typically expressed in the form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. In the function \( f(x) = 2x \), the constant \( m \) is the slope (2 in this case), and \( b \) is the y-intercept, which is 0 here since it's not present in the equation.
Linear functions graph as straight lines, making them very predictable and easy to work with. They describe relationships where there is a constant rate of change. This means that for every unit increase in \( x \), the function \( f(x) \) increases by a constant amount, specifically the slope \( m \).
This characteristic makes linear functions arguably one of the most important concepts in algebra. They serve as the foundation for understanding more complex functions and models later on.

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers that can be found on the number line. This set encompasses both rational numbers, like integers and fractions, as well as irrational numbers, such as \( \sqrt{2} \) and \( \pi \).
In simpler terms, real numbers are any numbers you can think of that don't involve imaginary components. Imaginary numbers are a separate category where numbers are multiplied by the imaginary unit, noted as \( i \), where \( i^2 = -1 \).

• Real numbers can be positive, negative, or zero.
• They are used in a vast array of scenarios across all areas of math.
• They form the basic set of values for most functions, including linear ones.

In the context of the given function \( f(x) = 2x \), the real numbers are significant because they define the domain of the function.

Function Domain

The domain of a function is simply the set of all possible input values, \( x \), that the function can accept without running into any undefined situations. Different types of functions can have different restrictions, but for a linear function like \( f(x) = 2x \), the domain is all real numbers.
Linear functions are continuous everywhere on the real number line because they don't have points where they become undefined. There are no divisions by zero and no square roots of negative numbers in linear equations. As a result, their domain can conveniently be written in interval notation as \((-finity, finity)\).
Understanding the domain is crucial for graphing functions, solving equations, and applying them in real-world scenarios. It's a basic but essential skill in mathematics that serves as a building block for many advanced topics.