Problem 123 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section. -Consider the function defined by $$\\{(-2,4),(-1,1),(1,1),(2,4)\\}$$ Reverse the components of each ordered pair and write the eresulting relation. Is this relation a function?

Step-by-Step Solution

Answer

After reversing the ordered pairs, we get the relation \[\{(4,-2),(1,-1),(1,1),(4,2)\}\]. But, this reversed relation is not a function, because one input (4) has more than one output.

Step 1: Understand the original Ordered Pair

The function listed, \[\{(-2,4),(-1,1),(1,1),(2,4)\}\], is a set of ordered pairs. The first element in each pair is the 'input' or 'x value', and the second element is the 'output' or 'y value'.

Step 2: Reverse the ordered pairs

To reverse the ordered pairs, it means to switch the input values with the output values so that the functions would look like this: \[\{(4,-2),(1,-1),(1,1),(4,2)\}\]

Step 3: Check if the reversed relation is a function

Now, we need to check if this reversed relation is still a function. Remember, a function is a relation where every input has exactly one output. In this case, the input '4' has two outputs, '-2' and '2'. Therefore, this reversed relation is not a function.

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is a fundamental concept used to organize data in a specific order. An ordered pair consists of two elements arranged as \((x, y)\). Here, \(x\) is typically called the input, and \(y\) is known as the output. The order of these elements is crucial since reversing them changes their meaning.
Ordered pairs are often used to represent coordinates on a graph where \(x\) denotes the horizontal placement and \(y\) shows the vertical position. In the context of functions, the first number in each pair represents the possible input values, while the second number represents the resulting output when you apply the function rule.

• Key Point: Changing the order of the elements creates a different meaning.
• Example: \((-2, 4)\) and \((4, -2)\) are different ordered pairs.
• Importance: Maintaining the correct order is crucial in analyzing functions and relations.

Relations

A relation in mathematics is a set of ordered pairs. It describes how elements from one set are related to elements from another set. A function is a specific type of relation where each input is related to exactly one output. But not all relations are functions.

• Relation: Can have multiple outputs for a single input.
• Function: Every input corresponds to one and only one output.

When you're given a relation, such as \(\{(-2,4),(-1,1),(1,1),(2,4)\}\), swapping the numbers in each pair (reversing) results in a new set, like \(\{(4,-2),(1,-1),(1,1),(4,2)\}\). This new collection of ordered pairs is again a relation.
To check if a relation is a function, you verify that no input is connected to more than one output. However, in this case, number '4' maps to both '-2' and '2', which means this reversed relation is not a function.

Inputs and Outputs

Inputs and outputs are integral to understanding functions and relations. They help calculate the correspondence between two sets of data. In functions, the 'input' is the value you provide to the function, often represented as \(x\). This input is processed according to a particular rule or operation, resulting in an 'output', which is represented as \(y\).

• Input: The starting value or the 'cause' in a function.
• Output: The result or the 'effect' after processing the input.
• Example: For the ordered pair \((-2,4)\), \(-2\) is the input and \(4\) is the output.

In the reversed relation \(\{(4,-2),(1,-1),(1,1),(4,2)\}\), the roles of inputs and outputs are switched. Hence, previously what were outputs become inputs and vice versa, showcasing a different scenario and set of relationships.