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When dealing with functions, understanding ordered pairs is essential. An ordered pair is simply a set of two numbers, usually written as \((x, y)\). The first number, \(x\), is the input, and the second number, \(y\), is the output. Think of an ordered pair as a pairing where each input has a specific output.
For instance, if you have the ordered pair \((2, 5)\), it means that when the input is 2, the output is 5.
Ordered pairs are often used to represent functions, especially in sets. For example, in the provided function \(f=\{(2,5),(3,9),(-1,11),(5,3)\}\), each pair links an input \(x\) to an output \(y\).

Evaluating a function means finding the output (or value) of a function for a specific input. When given a set of ordered pairs, this process involves locating the pair where the input matches the given value and then identifying the corresponding output.
Let’s look at how to evaluate the function for specific inputs using the given example function \(f=\{(2,5),(3,9),(-1,11),(5,3)\}\).

(a) Finding \(f(2)\)
To find \(f(2)\), we look for the pair where the first number is 2. From the set, this is the pair \((2, 5)\). Therefore, the value of \(f(2)\) is 5.
(b) Finding \(f(-1)\)
Similarly, to find \(f(-1)\), we look for the pair where the first number is -1. This is the pair \((-1, 11)\). Thus, the value of \(f(-1)\) is 11.

The input-output relationship in functions describes how each input corresponds to an output. This relationship is fundamental in understanding and working with functions.
A function can be seen as a machine or a rule that takes an input and produces an output. Each input has exactly one output in a function. This is crucial because it ensures consistency and reliability in calculations.
When looking at the set of ordered pairs \(f=\{(2,5),(3,9),(-1,11),(5,3)\}\), we can see this relationship clearly:

• Input 2 gives output 5.
• Input 3 gives output 9.
• Input -1 gives output 11.
• Input 5 gives output 3.

Understanding the input-output relationship helps in predicting the behavior of a function and efficiently finding the output for any given input in the function's domain.