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In mathematics, an ordered pair is a set of two elements. It follows the format \(x, f(x)\). Here, 'x' represents the input value, and 'f(x)' represents the output value, also known as the result. These pairs are called 'ordered' because the sequence in which the elements are written matters. For example, \(2, -5\) and \(-5, 2\) represent different pairs.

Ordered pairs are essential in understanding functions, as they clearly show the relationship between an input and its corresponding output. When given a function as a set of ordered pairs, you can determine the output just by looking at the input value. For instance, in the problem function \(\{(-1,-5),(0,5),(2,-5)\}\), each pair indicates what output (second element) corresponds to each input (first element).

Remember that to find a specific output, you simply need to locate the ordered pair with the corresponding input value.

Function notation is a mathematical system used to denote functions systematically. It's expressed in the form \(f(x)\), where 'f' is the function and 'x' is the input variable. This notation simplifies the representation of functions and makes it easier to handle complicated expressions.

For instance, if you have a function denoted by \(f\), and you want to find out what the function gives when you input 2, you write it as \(f(2)\). This concise form eliminates the need for lengthy descriptions and makes calculations straightforward.

In our exercise, the function is given by the set \(\{(-1,-5),(0,5),(2,-5)\}\). When asked to find \(f(2)\), it means we are looking for the output associated with the input value 2. Similarly, \(f(-1)\) means we're identifying the output when the input is -1. Understanding this notation is crucial for efficiently evaluating functions and solving problems.

Evaluating functions means finding the output value corresponding to a specific input value. This process often uses ordered pairs or function notation.

Let's break down the steps for evaluating a function using our example function \(\{(-1,-5),(0,5),(2,-5)\}\):

• Identify the input value given (e.g., 2 or -1).
• Locate the ordered pair containing this input in the function set.
• The second element of this pair is the output value.

For \(f(2)\): Look for the pair where the first element is 2. The pair is \(2, -5\), so \(f(2) = -5\).

For \(f(-1)\): Locate the pair where the first element is -1. The pair is \(-1, -5\), so \(f(-1) = -5\).

By following these steps, you can quickly find the outputs for any input values, making the function much easier to understand and utilize.