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Ordered pairs are a fundamental concept in mathematics, especially when dealing with graphing functions. An ordered pair is typically written as \((x, y)\), where \(x\) represents a value from the domain and \(y\) represents the corresponding function value, or output. These pairs provide a coordinate on a graph.

In the given exercise, we calculated function values for a specific discrete domain and represented these as ordered pairs. For example, the function value at \(x = 5\) is \(0\), resulting in the ordered pair \((5, 0)\). Similarly, each value of \(x\) corresponds to a function output, forming a set of points that can be graphed to visualize the function.

The order in these pairs is crucial: they indicate the position on the graph, with \(x\) as the horizontal position and \(y\) as the vertical position. Misplacing these values can lead to human errors when plotting them graphically. It's always \((x, f(x))\), not the other way around.

Graphing a function is all about visually representing mathematical relationships. By plotting ordered pairs on a graph, you create a clear picture of how the function behaves. This is especially useful for linear functions, like the one in our exercise. Linear functions produce straight lines when graphed because their rate of change, or slope, is constant.

To graph the function \(f(x) = -4 + 0.8x\), we use the ordered pairs derived from the discrete domain. Plotting these pairs \((5, 0), (6, 0.8), (7, 1.6), (8, 2.4), (9, 3.2), (10, 4)\) on a Cartesian coordinate grid results in a series of points.

Connecting these points with a straight line shows the linear nature of the function. This line illustrates how the function increases uniformly as \(x\) increases. Graphing helps in visualizing trends and makes it easier to predict future values and understand the overall function behavior.

Function values are essentially the outputs you get when you input certain values into a function. For linear functions such as \(f(x) = -4 + 0.8x\), calculating these involves substituting given \(x\) values from the domain into the function equation.

For example, when \(x = 5\), substituting this into the function gives:

• \(f(5) = -4 + 0.8 \times 5 = 0\)

This output, \(0\), is the function value for \(x = 5\). Continuing this method for each value in the discrete domain \(x = 5, 6, 7, 8, 9, 10\), you can determine all function values.

This step-by-step calculation helps ensure accuracy when forming ordered pairs. It's a simple yet systematic approach that provides a deeper understanding of how changes in \(x\) affect \(f(x)\).

A discrete domain in a function refers to specific, separate values that the input variable \(x\) can take. Unlike a continuous domain, where \(x\) can be any value in an interval, a discrete domain has distinct, individual values.

In this exercise, the domain was discrete, with \(x\) values specified as \(5, 6, 7, 8, 9, 10\). This means that we only calculated function values for these particular inputs.

Understanding the difference between discrete and continuous domains is important because it affects how we calculate and graph function values. A discrete domain often results in a scatter plot of individual points, while a continuous domain leads to a smooth and unbroken graph line.