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Chapter 5: Problem 4

Find the inverse of the relation. $$\\{(-1,3),(2,5),(-3,5),(2,0)\\}$$

Short Answer

Expert verified
The inverse relation is $$\{(3, -1), (5, 2), (5, -3), (0, 2)\}$$.

Step by step solution

01

- Understand the concept of inverse of a relation

The inverse of a relation is found by swapping the x-coordinates with the y-coordinates in each ordered pair.
02

- Swap coordinates for each pair

Given the relation $$\{(-1,3),(2,5),(-3,5),(2,0)\},$$ we need to swap the coordinates of each pair:1. \((-1, 3) \rightarrow (3, -1)\)2. \((2, 5) \rightarrow (5, 2)\)3. \((-3, 5) \rightarrow (5, -3)\)4. \((2, 0) \rightarrow (0, 2)\)
03

- Write the inverse relation

The inverse relation, after swapping coordinates, is: $$\{(3, -1), (5, 2), (5, -3), (0, 2)\}$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ordered pairs
In mathematics, an ordered pair is a pair of elements written in a specific order. It is usually written in the form \( (x, y) \). The first element is the x-coordinate, and the second element is the y-coordinate. These pairs are used in coordinate geometry to represent points on a plane.

An ordered pair allows us to find and plot the location of a point on the Cartesian plane.
When dealing with relations, it is common to have a set of multiple ordered pairs.
  • \( (-1, 3) \)
  • \( (2, 5) \)
  • \( (-3, 5) \)
  • \( (2, 0) \)
In this example, each pair represents a specific point with a unique x (horizontal) and y (vertical) value.
coordinate swapping
Coordinate swapping is a key concept when finding the inverse of a relation. It involves exchanging the positions of the x-coordinate and the y-coordinate in each ordered pair.

This process transforms \( (x, y) \) into \( (y, x) \).By doing so, we essentially 'flip' each point over the line \( y = x \).This operation is straightforward but crucial for converting a relation to its inverse.
Let's apply coordinate swapping to the example from the exercise:
  • \( (-1, 3) \rightarrow (3, -1) \)
  • \( (2, 5) \rightarrow (5, 2) \)
  • \( (-3, 5) \rightarrow (5, -3) \)
  • \( (2, 0) \rightarrow (0, 2) \)
After swapping coordinates, the new set of ordered pairs forms the inverse relation.
inverse function
An inverse function essentially reverses the roles of the input and output of a given function or relation. For a function \( f(x) \), the inverse is denoted as \( f^{-1}(x) \). This new function maps \( f(x) \) back to \( x \).

In simpler terms, if the function \( f \) takes \( x \to y \), then the inverse function \( f^{-1} \) takes \( y \to x \).
For instance, given a relation consisting of ordered pairs:
  • \( (-1, 3) \rightarrow (3, -1) \)
  • \( (2, 5) \rightarrow (5, 2) \)
  • \( (-3, 5) \rightarrow (5, -3) \)
  • \( (2, 0) \rightarrow (0, 2) \)
This swapping leads to the inverse relation, which reverses the original relationship between variables.

Understanding inverse functions is fundamental in various mathematical applications, including solving equations and transforming graphs.

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