91Ó°ÊÓ

Absolute value functions are a fun and vital part of calculus. They tell us how far a number is from zero, regardless of its direction on the number line. The absolute value, denoted with vertical bars like \(|x|\), is determined by whether the value of \(x\) is positive, negative, or zero.

• If \(x\) is positive or zero, then \(|x| = x\).
• If \(x\) is negative, then \(|x| = -x\), because we take the opposite to make it positive.

Absolute value functions are continuous everywhere on the number line. They don't break or skip any numbers, which is why, even if \(x\) moves from positive to negative, the absolute value remains seamless. For instance, in the function \(f(x) = \frac{1}{2}|2x|\), the absolute value ensures there is no disruption as \(x\) crosses zero.

Piecewise functions might look complicated, but they're just like a patchwork quilt. They are made up of different "pieces," where each "piece" applies to a specific part of the domain.

These types of functions allow us to define various behaviors over different intervals, which can be useful for modeling real-world situations that have different conditions.
For example:

• A function could be defined as \(x^2\) for \(x < 0\) and \(-x^2\) for \(x \geq 0\).

For a piecewise function to be continuous, it must connect smoothly at the boundaries where the pieces meet. This means no breaks or jumps at those transition points. In the case of our function \(f(x) = \frac{1}{2}|2x|\), it's essentially piecewise because it can be split at 0: when \(x < 0\), the function is linear, and so is it when \(x \geq 0\). But since both parts are seamlessly connected, it's continuous everywhere.

Interval notation is a handy mathematical shorthand used to express where functions are defined or continuous. Instead of listing every single value, we use intervals, where:

• Parentheses \(()\) indicate that an endpoint is not included.
• Brackets \([]\) mean the endpoint is included.

This is particularly useful in calculus and can simplify expressing domains and ranges. In our given function \(f(x) = \frac{1}{2}|2x|\), we stated it's continuous across all real numbers.

So, in interval notation, we write this as \((-∞, ∞)\). This notation means the function has no gaps or jumps from negative to positive infinity, allowing for easy understanding and clear communication of which values make the function continuous.