91Ó°ÊÓ

In mathematics, the concepts of domain and range are fundamental in understanding functions. The domain of a function is the complete set of possible values of the independent variable, which is typically represented by 'x'.
The range, on the other hand, consists of all possible values of the dependent variable (often represented by 'y') that the function can produce.

For example, consider the function given in the exercise: \( f(x) = 2x \). To find the domain, we ask ourselves: For which values of x is the function defined? Since the function is linear, it is defined for all real numbers. Therefore, the domain is \((-fty, fty)\).

Similarly, to find the range, we consider all possible values the function can output. As it's a linear function without any restrictions, the range is also all real numbers \((-fty, fty)\).
This understanding is crucial when graphing and analyzing various functions, as it provides insight into the behavior and limitations of the function.

Interval notation is a concise way of expressing the domain and range of functions. Rather than listing all possible values, interval notation captures the range of values in brackets.

For example, for the domain of \( f(x) = 2x \), because x can be any real number, we use the notation \((-fty, fty)\). This indicates that the function is defined from negative infinity to positive infinity.
The same goes for the range: the function can produce any real number, which we also write as \((-fty, fty)\).

Important elements to remember about interval notation:

• Parentheses \(( )\) indicate that the endpoint is not included.
• Square brackets \([ ]\) indicate that the endpoint is included.
• \(-fty\) and \(fty\) are always written with parentheses because they represent unbounded limits.

Using interval notation simplifies the process of documenting and communicating the scope of a function's domain and range.

Plotting points is a foundational skill in graphing functions. It involves selecting various x-values, substituting them into the function to determine their corresponding y-values, and then placing the resulting points on a coordinate plane.

For the function \( f(x) = 2x \), let's select a few values of x and find their corresponding y-values:

• When \( x = -2 \), then \( f(x) = 2(-2) = -4 \)
• When \( x = 0 \), then \( f(x) = 2(0) = 0 \)
• When \( x = 2 \), then \( f(x) = 2(2) = 4 \)

These points can be written as coordinates: \((-2, -4)\), \((0, 0)\), \((2, 4)\).

Next, plot these points on a coordinate plane. Once the points are placed, connect them with a straight line, as the given function is linear (having a constant rate of change). This graphical representation helps visualize the function's behavior, confirming the understanding of its domain and range.
With practice, plotting points becomes intuitive, allowing you to graph more complex functions accurately.