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The concept of ordered pairs is a fundamental aspect of understanding functions in mathematics. An ordered pair consists of two elements, where the order in which these elements appear is crucial. The standard notation for an ordered pair is \(a, b\), where \(a\) is the first element and \(b\) is the second element.

In the context of functions, ordered pairs represent input-output relationships. The first element of each pair is an input, typically known as the x-value or domain element, and the second element is the output, or the y-value, resulting from the function acting on the input.

Importance in Function Analysis

When we look at a set of ordered pairs that defines a function, we can determine the function's domain and range quite easily. The domain is the set of all first elements, and the range is the set of all second elements. In our exercise example, the function \(h\) consists of the ordered pairs \((10,100)\), \((20,200)\), \((30,300)\), and \((40,400)\), illustrating the input-output correlation distinctly.

The representation of a function can be visualized in various forms, including ordered pairs, graphs, tables, equations, or mappings. Each of these representations holds the same underlying concept: a function assigns exactly one output to each input within its domain.

In this exercise, we are given a function \(h\) represented by a set of ordered pairs. This is a discrete representation, meaning the function is only defined for certain specific inputs.

What does this tell us?

The clearest outcome of analyzing a function through its ordered pairs is being able to identify its domain and range immediately, as each input and corresponding output is explicitly stated. It is paramount to recognize that each input in this form of representation has a unique output, which is a defining characteristic of a function.

In elementary algebra, understanding the behavior of functions is key, and this often involves examining their domain and range. The domain of a function includes all the possible inputs, while the range consists of all the possible outputs that the function can produce.

To find the domain and range, we list the unique values that appear as the inputs and outputs in the function's representation. With the function \(h\) in our exercise, recognizing the domain \(\{10,20,30,40\}\) and the range \(\{100,200,300,400\}\) is a matter of extracting the corresponding elements from the ordered pairs.

Connecting Concepts

Through this step-by-step approach, we apply the principles of elementary algebra to analyze the function systematically. The careful extraction of these sets is an excellent example of algebraic thinking, bridging the gap between abstract concepts and practical application.