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Precalculus is a course that prepares students for the complexities of calculus, and one fundamental concept covered is the understanding of intervals on a number line. An interval is a set containing all the numbers between two endpoints. It can be visualized as a continuous section of the number line and is often used to express the domain or range of a function.

For instance, the interval \( (-\infty, 4) \) includes all real numbers less than 4, but does not include 4 itself. We can recognize this because the parenthesis \( ( \) denotes exclusion. Similarly, the interval \( (-2,6] \) includes all numbers from -2 up to and including 6, as illustrated by the use of the square bracket \( ] \) at the right end. Understanding and combining these intervals is a pivotal skill that allows students to solve complex problems in calculus.

Set theory is the branch of mathematical logic that studies collections of objects, known as sets. In this context, intervals are considered sets of numbers with specific properties that define which numbers are included in the interval or set.

In the exercise given, we have two sets described as intervals. The operation we are concerned with is the union, denoted by \( \cup \), which combines all the elements (numbers) from the two intervals into a single set so that no number is counted twice. When executing a union operation on intervals that have overlapping parts, we merge them to reflect a set that contains all the values from both intervals without duplication.

Additionally, understanding the fundamental operations in set theory such as union, intersection, and set difference is vital. These operations allow students to work with multiple sets and find commonalities or unique elements, skills that are useful across many areas of mathematics and related fields.

The union of intervals is a concept derived from set theory, applied to the particular case of numerical intervals. When we say union, symbolized by \( \cup \), we are referring to the combination of all elements from two or more intervals. If the intervals overlap, the union simplifies to a single, more extensive interval that encompasses all the values from the overlapping intervals.

Looking at the provided exercise, it's clear that the two intervals \( (-\infty, 4) \) and \( (-2,6] \) have common numbers. Particularly, every number between -2 and 4 is included in both. The solution process involves visualizing these intervals, determining the overlap, and then writing the union as the smallest interval that encases all numbers of the original intervals. Here, that resultant interval is \( (-\infty, 6] \), a classic example of a union of two intervals that simplifies the representation of a set of numbers.