Problem 38 - Calculus: One and Several Variables

Question

Give the domain and range of the function. $$f(x)=|2 x+1|$$

Step-by-Step Solution

Answer

The domain of the function \(f(x)=|2x+1|\) is all real numbers (\(-\infty, \infty\)) and the range of the function is all non-negative real numbers (\([0, \infty)\)).

Step 1: Find the Domain of the Function

For the given function \(f(x) = |2x+1|\), it is defined for all real numbers since the absolute value of any real number is a real number. Therefore, the domain of this function is all real numbers. In interval notation, it is \(-\infty, \infty\).

Step 2: Find the Range of the Function

Since the absolute value of any real number is always non-negative, the range of the function is all non-negative real numbers. In interval notation, it is \([0, \infty)\).

Key Concepts

Absolute Value Function

The absolute value function is a fascinating mathematical concept that measures the distance of a number from zero on the number line, regardless of direction. This means when you apply the absolute value operation to a number, it always returns the non-negative form of that number. For example, the absolute value of both \(-5\) and \(5\) is the same, which is \(5\).
When we have a function like \(f(x) = |2x + 1|\), it means we're interested in how the expression \(2x + 1\) behaves when its absolute value is considered. This particular function transforms any value of \(x\) first by multiplying by \(2\), then adding \(1\), and finally taking the absolute value of the result.

• It's crucial to note that absolute value functions are always defined for all real numbers, as you can always calculate the distance from zero for any real number.
• They tend to create a distinctive "V" shape graph, which is symmetric around the y-axis if expressed as \(f(x) = |x|\).
• When the expression inside the absolute value includes a variable term (like our \(2x + 1\)), it affects the location of the "V" on the graph.

Interval Notation

Interval notation is a convenient mathematical shorthand to describe sets of numbers between two endpoints. It is primarily used to state the domain and range of functions in a concise manner. The key symbols used in interval notation include:

• Parentheses \((\) and \()\) for open intervals, where the endpoints are not included in the set.
• Brackets \([\) and \()]\) for closed intervals, where the endpoints are included.
• A comma \(,\) separates the two endpoint values.
• For unbounded intervals, such as those extending indefinitely, infinity symbols \(\infty\) or \(-\infty\) are used.

For example, the domain of the function \(f(x) = |2x+1|\) can be represented in interval notation as \((-\infty, \infty)\), indicating that it accepts all real numbers. Its range, another important aspect, is expressed as \([0, \infty)\), reflecting that the output is always zero or a positive number. Understanding interval notation is essential for efficiently communicating the set of values a function can accept or produce.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes both rational numbers such as fractions and integers, and irrational numbers, which cannot be written exactly as fractions (like \(\pi\) or the square root of \(2\)). Real numbers are incredibly important because they form the backbone of most of classical mathematics.

• The domain of the function \(f(x) = |2x+1|\) being all real numbers highlights this aspect, because the function is defined everywhere along the number line.
• Real numbers are used to describe both the possible input values (domain) and the set of possible outcomes (range) for a function.
• In this function, while \(x\) can be any real number, because of the absolute value, the outputs are restricted to non-negative real numbers, hence the range of \([0, \infty)\).

Grasping the concept of real numbers helps in understanding how functions interact with the number system we most commonly use.