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High School Mathematics
8.5 Inverse Relation or Inverse Functions

Inverse Relation:
If R is a relation from set A into another set B, then by interchanging the first and second coordinates or ordered pairs of R we get a new relation. This relation is called the inverse relation of R and is denoted by R-|.
Set builder form,
R-| = {(y,x)/(x,y) R}.

Note:
Domain and range of R-| are respectively range and domin of R.

Inverse Functions:
To find the inverse of a function written as an equation, interchange the two variables x and y and solve for y. If the inverse is also a function, it is denoted by f-1

Example 1:
Find the inverse of the function {(3,4),(5,6),(7,8),(9,10)}. State the domain and range of this inverse. State if the inverse is also a function.
Original function: {(3,4),(5,6),(7,8),(9,10)}
Interchange the first and second coordinates in each pair.
Inverse of the function: {(4,3),(6,5),(8,7),(10,9)}
domain: {4,6,8,10}
range: {3,5,7,9}
Since each element in the domain of the inverse maps onto one and only one element in the range, the inverse of the original function is also a function.

Example 2:
Find the inverse of the function y = 2x + 4.
Original function: y = 2x + 4
Domain of f: {x: x is real}
Range of f: {y: y is real}
To find the inverse of the function interchange x and y
x = 2y + 4
x/2 - 4 = 2y
x/2 - 2 = y
Therefore the inverse function f-1 is: y = x/2 - 2
Domain of f-1: {x: x is real}
Range of f-1: {y: y is real}


Directions: Choose the correct answer. Also write at least ten examples of your own.
Q 1: If R = {(1,2),(2,3),(3,4),(4,5)} is a relation then find its inverse relation.
{(1,2),(2,3),(3,4),(4,5)}
{(2,1),(3,2),(4,3),(5,4)}

Q 2: Find the inverse of the function:
y = 2x - 5, x = -1, -2, -3
y = x/2 + 5/2; x = -7, -9, -11
y = x + 5; x = 3, 5, 7
x = y/2 + 5/2; x = 7, 9, 11

Q 3: If R = {(1,3),(2,5),(3,7),(4,9)} is a relation then find its inverse relation.
{(3,1),(5,2),(7,3),(9,4)}
{(1,3),(2,5),(3,7),(4,9)}

Q 4: Find the inverse of the function:
{(1,3),(2,3),(4,5),(9,5)}
{(1,3),(2,3),(4,5),(9,5)}
{(3,1),(3,2),(5,4),(5,9)}
Not a function

Q 5: If R-| = {(1,2),(3,4),(5,6)} is a relation then find its range.
{1,3,5}
{2,4,6}

Q 6: Find the inverse of relation or function: y = 4x - 1
x = 4y + 1
x = 4/y - 1/y
y = x/4 + 1/4

Q 7: If R = {(1,2),(3,4),(5,6)} is a relation then find the range of R-|.
{2,4,6}
{1,3,5}

Q 8: Find the inverse of the function
f(x) = -x + 7
y = 7- x
x = 7 - y
x - y = 7

Question 9: This question is available to subscribers only!

Question 10: This question is available to subscribers only!


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