Find The Inverse of a Relation - Questions With Solutions

How to find the inverse of a relation given by its graph? Examples are presented along with detailed solutions and aslo questions with Solutions and explanations are included.

Examples: Find the Inverse of a Relation Given by its Graph

A point (ordered pair) on the graph of the inverse of a relation is obtained by interchanging (switching) the coordinates of a point (ordered pair) in the graph of the relation. Geometrically, a point $(a,b)$ on the graph of a relation and a corresponding point $(b,a)$ on the inverse of the relation are reflection of each other on the line $y=x$ .

Example 1

Step 1: Select points on the graph of the given relation and find their coordinates $(a,b)$. The selected points are marked in blue on the graph below with the following coordinates:

$(8,4),(5,2),(4,0),(3,-1),(0,-2),(-2,-4),(-3,-5)$

Step 2: Reflect the points on the line $y=x$ by switching the $x$ and $y$ coordinates, The reflected (red) points : $(b.a)$ are:

$(4,8),(2,5),(0,4),(-1,3),(-2,0),(-4,-2),(-5,-3)$

Sketch the graph of the inverse of the given relation by connecting the reflected points shown in red. The original graph and its inverse are mirror images of each other across the line $y=x$.

Example 2

Solution:

Step 1: Select and define points on the graph of the given relation by their coordinates $(a,b)$ as shown on the graph below (blue points on the graph).

The selected points are: $(6,3),(2,-5),(0,-1),(-2,-5),(-3,-3)$

Step 2: Reflect the points across the line $y=x$ by switching the $x$- and $y$-coordinates, as shown by the red points with coordinates $(b,a)$.

Reflected points: $(3,6),(-5,2),(-1,0),(-5,-2),(-3,-3)$

The graph of the inverse relation is obtained by connecting the reflected points. The original graph and its inverse are mirror images of each other across the line $y=x$.

Questions

Sketch the graph of the inverse of each of the relations given by its graph below:
a)

Solutions to the Above Questions

Step 1: Select points on the graph of the given relation and find their coordinates: blue points shown on the graph below with the following coordinates:

$(4,2),(2,2),(1,-1),(0,-2),(-1,-1),(-2,-4)$

Step 2: Reflect on the line $y=x$ by switching the $x$ and $y$ coordinates of the points obtained above to get the red points:

$(2,4),(2,2),(-1,1),(-2,0),(-1,-1),(-4,-2)$

Step 1: Select points on the graph of the given relation and find their coordinates. Blue points are shown on the graph below with the following coordinates:

Step 2: Reflect the points (by switching the $x$ and $y$ coordinates) across the line $y=x$. These are shown as red points:

More References and links

• Inverse Function
• Find Inverse Function (1) - Tutorial
• Find Inverse Function (2) - Tutorial
• Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
• High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers
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