Problem 93 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$f(x)=|x|$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = |x|\) is \(\boxed{(-\infty, \infty)}\), as the absolute value function is defined for all real numbers.

Step 1: Analyzing the function given

The function is defined as \(f(x) = |x|\), which represents the absolute value of x. The absolute value function returns the distance of any real number x from 0, and it is always a non-negative number. Since this function is defined for all real numbers, there are no restrictions on the values of x that can be used as inputs.

Step 2: Determine the domain of the function

Since the absolute value function is defined for all real numbers, the domain of the given function \(f(x) = |x|\) is the set of all real numbers. In interval notation, we can express the domain as \((-\infty, \infty)\). So, the domain of the function \(f(x) = |x|\) is: \(\boxed{(-\infty, \infty)}\)

Key Concepts

Domain of a Function

When we talk about the "domain" of a function, we're referring to all the possible input values that the function can accept without running into any issues. In simpler terms, it's the set of all x-values that you can plug into the function and get a valid y-value back.
For the function given, which is the absolute value function \(f(x) = |x|\), it turns out to be pretty straightforward. The absolute value of x, \(|x|\), is defined for every real number. Why? Because every real number has a distance from zero, and the absolute value just measures that distance.
Hence, the domain of the absolute value function includes every possible real number.

• For example, you can plug in negative numbers, positive numbers, or even zero into \(f(x) = |x|\), and you'll always get a real number back.
• There's no division by zero or taking the square root of a negative number to worry about in this case.

Interval Notation

After finding out that a function is defined for a certain range of numbers, we often need a concise way to express these ranges. That's where "interval notation" comes in. It's a shorthand used in mathematics to denote which numbers are included in a set.
For the absolute value function \(f(x) = |x|\), we determined that its domain is all real numbers. In interval notation, we express this as \((-\infty, \infty)\).
You might be wondering what these symbols mean:

• The parenthesis \((\) and \()\) indicate that the endpoints \(-\infty\) and \(\infty\) are not included, as infinity isn't a number you can "reach." It just signifies that the numbers go on forever in that direction.
• This notation is a neat way to show that all numbers from negative infinity to positive infinity are part of the set.

In interval notation, brackets \([\) or \()]\) would be used to include endpoints if the interval had been closed, like \([1, 5]\), including both 1 and 5. But here, since real numbers extend infinitely, we use parentheses.

Real Numbers

Real numbers are the backbone of most of the numbers we use in mathematics. They include all the rational numbers, such as fractions and whole numbers, and all the irrational numbers, numbers that can't be written as a simple fraction.
Examples of real numbers include integers like -3, zero, and 5, rational numbers like \( \frac{1}{2} \) or 3.14, as well as irrational numbers like \(\pi\) or the square root of 2. In essence, if you can locate it on a number line, it's a real number!
Therefore:

• The function \(f(x) = |x|\), which is the absolute value function, applies to any real number. That's because any real number has a corresponding point on a number line and hence has a definite distance from 0.
• This inclusion of all real numbers is why the domain of \(f(x)=|x|\) is expressed in interval notation as \((-\infty, \infty)\).

Every real number is a valid input for our absolute value function, demonstrating its broad applicability and versatility in various mathematical scenarios.