Lesson 2: Inverse of Relations and Their Graph
Lesson Objective:
Dear learner,
by the end of this lesson you will be able to:
- Define inverse of a relation.
- Determine domain and range of inverse of relations.
- Draw graph of relations and their inverse.
- Solve problems given related to inverse of relations.
Key Terms
- Relation
- Inverse of a relation
- Graph of Inverse relations
Brainstorming Activity
Consider the given relation $R = \ \{(2,3),(4,5),(7,8),(8,9)\}$. define an other relation H by interchanging the components of ordered pairs of R and list out domain and range of H?
I think you try to solve the problem by your own, first detrmine H and set the second componenet on the place of the first component and the first component on the place of second component sothat $H =\ \{(3.2),(5,4),(8,7),(9.8)\}$ is an other relation.
H = $R^{-1} = \ \{(3.2),(5,4),(8,7),(9.8)\}$
- Domain $R^{-1}$ = {3,5,8,9} which is the same as Range of R
- Range $R^{-1}$ = {2,4,7,8} is the same as Domain of R
Inverse of a Relation:
Definition: the inverse of relation R denoted by $R^{-1}$ is a relation obtained by interchanging components of ordered pairs of R.
i.e.$ R^{-1} = \ \{(x,y): (x,y) \in R\}$
Example 1:
Find the inverse, domain of $R^{-1}$ and range of $R^{-1}$ for the following relations
A). Given the relation $R$ = {(-3,4),(6,12),(8,25)}
B). R= {(x,y): x is a mother of y}
C). $ R=\ \{(x,y): x,y \in \mathbb{R}, y = 2x-5\}$
D). $R = \ \{(x,y): x,y \in \mathbb{R}, y < 3x$ and $ y \ge 2x+1\}$
Solution:
General: Steps to find $R^{-1}$
- Interchanging coordinates is enough or
- After interching variables solve y interms of x
- Fix $R^{-1}$
A). Interchanging coordinates gives $R^{-1} = \ \{(4,-3), (12,6),(25,8)\}$
- Domain $R^{-1}$ = {4,12,25} = Range of R and range $R^{-1}$ = {-3,6,8} = Domain of R
B). $R^{-1}$ can be written as $R^{-1} = \ \{(y,x): x is a mother of y\}$ or $R^{-1} = \ \{(x.y): y is a mother of x\}$
C). $R^{-1}$ can be simply written as $R^{-1}=\ \{(y,x): x,y \in \mathbb{R}, y = 2x-5\}$ or
first interchange variables of x and y as : $x = 2y-5$ and solve for y you get $ y = \frac{x+5}{2}$, then
$R^{-1} = {(x,y): x,y \in \mathbb{R}$ and $y = \frac{x+5}{2}\}$
- Domain $R^{-1} = \ \{x: x \in \mathbb{R}\}$
- Range $R^{-1} = \ \{y:y \in \mathbb{R}\}$
D). $R^{-1}$ can be written as:
$<span id="MathJax-Element-29-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><merror><mtext>R^{-1} = \ \{(x,y): x,y \in \matbb{R}, x < 3y and x \ge 2y+1}</mtext></merror>$
=$ \ \{(x,y): x,y \in \mathbb{R, y >\frac{x}{3}$ and $y \le \frac{x-1}{2}}$
therefore $<span id="MathJax-Element-31-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><merror><mtext>R^{-1} = \ \{(x,y): x,y \in \matbb{R, y >\frac{x}{3} and y \le \frac{x-1}{2}\}</mtext></merror>$R^{-1} = \ \{(x,y): x,y \in \matbb{R, y >\frac{x}{3} and y \le \frac{x-1}{2}\}
In general
- Domain of $R^{-1}$ = Range of R
- Range of $R^{-1}$ = Domain of R
Graph of Inverse Relations
Graphs of R and $R^{-1}$ are mirror images of each other under the line y = x.
Example 2:
Find the inverse of the following relations and draw graph of R and $R^{-1}$?
- a. R = {(1,3),(2,4),(3,3)}
- b. $R=\ \{(x,y): x,y \in \mathbb{R}, y = 2x-5\}$
- c. $R = \ \{(x,y): x,y \in \mathbb{R}, y < x+2 and y \ge x+1\}$
Solution:
a. the inverse of R is denoted by :$<span id="MathJax-Element-38-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><merror><mtext>R^{-1} = \ \ {(3,1),(4,2),(3,3)\}</mtext></merror>$R^{-1} = \ \ {(3,1),(4,2),(3,3)\}
Graph of R and $R^{-1}$ looks as drawn below:
b. $R^{-1} = \ \{(x,y): x,y \in \mathbb{R}, y = \frac{x+5}{2}\}$
The graph of R and $R^{-1} $ are drawn on the same coordinate plane as given below so that you can see they are mirror images of each other along the line y = x.
fig2.1: graph of R and $R^{-1}$
c.$R^{-1} = \ \{(x,y): x,y \in \mathbb{R}, y > x-2 and y \le x-1\}$
First Draw Graph of R and reflect it with respect to the line y =x you get graph of $R^{-1}$ as shown on fig 2.2:
fig. 2.2 graph of $ y < x+2 and y \ge x+1\}$ and $y > x-2 and y \le x-1\}$