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Linear functions are foundational concepts in algebra and essential for understanding various mathematical and real-world applications. At its core, a linear function is one that, when graphed, produces a straight line. The general form for a linear function is expressed as
\(y = mx + b\),
where \(m\) represents the slope of the line, which indicates how steep the line is. A positive slope means the line is inclined upward, while a negative slope means it's inclined downward. The \(b\) signifies the y-intercept, which is the point at which the line crosses the y-axis.
In the context of the given exercise, \(y = 21 - 2x\), the function represents a straight line with a negative slope of -2, meaning the line descends as it moves from left to right across the graph. The y-intercept is 21, which is where the line crosses the y-axis. This function is simple yet powerful, allowing you to predict the value of \(y\) for any given \(x\).

The domain of a function is the set of all possible input values (often represented by \(x\)) for which the function is defined, or put simply, the 'allowed' values that can go into a function. It is crucial because it addresses which numbers make sense for a specific mathematical rule. In many cases, particularly with linear functions, the domain can be all real numbers since there's no mathematical operation that would cause a problem for any real number input.
However, the domain can be limited for various reasons such as the context of a problem. For example, in our exercise, the domain provided is the set {0, 1, 2, 3}. This restriction could come from the scenario you are modeling, like only being open for business four days or having just four objects to count. Hence, when crafting an input-output table, we carefully match each domain value with its corresponding output, ensuring we have a clear picture of the function's behavior over the specified interval.

Function notation is the way in which we represent a function in algebra. It is a succinct method to write out the mathematical operations that will be performed on an input value to produce an output. Using the notation \(f(x)\) instead of \(y\) makes it extremely clear that the value obtained depends on the input \(x\).
For instance, our function could also be written as \(f(x) = 21 - 2x\). If you want to specify the value of the function when \(x = 1\), you would write \(f(1) = 19\). This notation is powerful because it facilitates the clear communication of the relationship between variables and is fundamental in more complex mathematical concepts. Equipped with function notation, you're well-prepared to build input-output tables, analyze function behaviors, and more.