91Ó°ÊÓ

In mathematics, a relation involves pairing elements from one set with elements of another set. Consider a set of inputs and a set of outputs. The relation is essentially the connection that links every input to an output. For example, if we have a relation defined by the pairs \( (-12,5), (-10,3), (8,3) \), the numbers on the left (like -12, -10, and 8) are the inputs, and the numbers on the right (like 5 and 3) are the outputs. This suggests that the input -12 is connected to the output 5, and so on.
It’s essential to understand that a relation can have multiple pairs, and sometimes the same output can be connected to different inputs. But it's the uniqueness of these connections that help us define whether a relation is a function, which brings us to our next concept.

Input-output pairs are at the heart of understanding relations and functions. Each pair connects an element from the domain (the set of possible inputs) to an element of the range (the set of possible outputs). By examining the input-output pairs given in the set \( (-12,5), (-10,3), (8,3) \), we identify:

• The inputs are -12, -10, and 8.
• The outputs are 5 and 3.

Here, the domain values (-12, -10, 8) are paired with the range values (5, 3). This is essential because to determine if a relation is a function, you need to ensure that each input (from the domain) connects to exactly one output (from the range).

One of the key characteristics of a function is that each input must map to one and only one output. This means there can be no input that maps to multiple outputs. To determine whether our given relation \( \{(-12, 5), (-10, 3), (8, 3)\} \) is a function, we need to confirm that every x-value (input) corresponds to just one y-value (output).
Let's break it down:

• The input -12 maps to the output 5.
• The input -10 maps to the output 3.
• The input 8 maps to the output 3.

Each input is paired with only one output, confirming that the relation functions as a valid function. Thus, our relation meets the criteria of having a unique output for every individual input, which is why it defines a function.