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College algebra serves as a foundations course for many higher-level mathematics classes and various fields requiring mathematical prowess. It covers a plethora of topics, with one of the fundamental concepts being the domain of a function.

The domain of a function, simply put, is the complete set of possible values of the independent variable, typically represented by 'x', for which the function is defined. Understanding the domain is crucial, as it informs us about the function's applicability and limitations. In the given exercise, we examine a linear function to determine its domain. Since linear functions do not have restrictions such as square roots of negative numbers or division by zero, the domain in this case consists of all real numbers, which is a notable characteristic of linear functions.

The linear function is a cornerstone concept of algebra, frequently represented in the form of a straight line when graphed on a coordinate plane. Governed by the formula of the type \(y = mx + c\), a linear function is characterized by a constant rate of change, which is the slope 'm'. The 'c' in the equation stands for the y-intercept, the point where the line crosses the y-axis.

In our original exercise, the function \(f(x) = 3(x - 4)\) is a linear function with a slope of 3 and a y-intercept at \((-4, 0)\). Unlike quadratic functions or rational functions, linear functions have no restrictions such as square roots or denominators that can be zero; hence the domain is all real numbers. This simplicity allows these functions to be highly applicable across various disciplines, including economics, science, and engineering.

Interval notation is a concise way of writing sets of numbers, generally used for expressing domains and ranges of functions. This mathematical shorthand is particularly helpful in denoting domains because it effectively describes intervals on the number line where the function exists without listing each individual element.

Here’s how it works: The use of parentheses, \((\) and \()\), signifies that an endpoint is not included in the interval, known as an 'open interval'. Conversely, brackets, \([\) and \(]\), denote that an endpoint is included, or a 'closed interval'. An infinite interval is represented by using an infinity symbol, and since infinity is a concept, not a number, it is always accompanied by a parenthesis. For example, in our solution, the domain of the linear function is all real numbers, which we write in interval notation as \((-\infty, +\infty)\). This indicates that there’s no restricting boundary for the x-values—it continues indefinitely in both directions on the number line.