Problem 37 - 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, \((- fty, \infty)\).

Step 1: Understanding Domain

The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical issue, such as division by zero or taking the square root of a negative number. For the function we have, \(f(x) = 2x\), it's a linear function.

Step 2: Analyzing the Function

In the function \(f(x) = 2x\), there is no operation that would restrict the input values; multiplication by 2 can be performed on any real number without causing any mathematical difficulties.

Step 3: Determining the Domain

Since there are no restrictions like division or square roots of negative numbers, every real number is a valid input for \(x\). Therefore, the domain is all real numbers.

Step 4: Expressing the Domain

The domain of the function \(f(x) = 2x\) is expressed in set notation as \(\{x \in \mathbb{R} \}\) or in interval notation as \((-fty, \infty)\), which indicates all real numbers.

Key Concepts

Linear Function

A linear function is a simple yet powerful type of function that forms the basis for more complex mathematical concepts. It takes the form of \(f(x) = mx + b\), where \(m\) and \(b\) are constants.

• The constant \(m\) represents the slope of the line, determining its steepness.
• The constant \(b\) stands for the y-intercept, indicating where the line crosses the y-axis.

A unique characteristic of linear functions is that their graphs are straight lines. These functions are widely used to model relationships where the rate of change is constant. For instance, the function \(f(x) = 2x\) from the exercise is a linear function with a slope of 2 and a y-intercept of 0. Because linear functions have no restrictions like division by zero or square root of negatives, their domains usually cover all real numbers.

Real Numbers

Real numbers include both rational and irrational numbers, making them very comprehensive and versatile in math.

• Rational numbers are those that can be expressed as a fraction, including integers and finite decimals.
• Irrational numbers cannot be expressed as simple fractions and include numbers like \(\pi\) and the square root of 2.

The concept of real numbers is crucial in understanding domains because when we say the domain is all real numbers, we mean all numbers that don't include any imaginary components. For a linear function such as \(f(x) = 2x\), the real numbers as domain mean you can input any number from this broad set without issue.

Set Notation

Set notation is a mathematical way to define collections of objects or numbers. It is often used to express domains and ranges neatly.

• A typical way to use set notation for domains is with curly braces and a description of the variable, for example, \(\{x \in \mathbb{R}\}\).
• \(\mathbb{R}\) symbolizes the set of all real numbers, so \(\{x \in \mathbb{R}\}\) means inclusive of every real number.

Set notation provides a clear and concise way to communicate the domain of functions, helping us avoid lengthy explanations. In the exercise, the domain of \(f(x) = 2x\) is described using the set notation \(\{x \in \mathbb{R}\}\), illustrating that any real number \(x\) is a valid input.

Interval Notation

While set notation is one way to define a domain, interval notation offers another versatile format. It involves writing the domain as a pair of numbers enclosed in brackets or parentheses, showing the range of permissible inputs.

• Parentheses \((-\infty, \infty)\) indicate that the interval is open, meaning it doesn't include the endpoints. Since \(-\infty\) and \(\infty\) are not actual numbers, they're never included.
• Brackets \([a, b]\) suggest a closed interval, where \(a\) and \(b\) are part of the set.

In the context of the exercise's linear function \(f(x) = 2x\), the domain is written as \((-\infty, \infty)\). This indicates every real number is part of the domain, seamlessly covering all possibilities from negative to positive infinity.