Inverse Relation - Formula, Graph | Inverse Relation Theorem

Example 1: Find the inverse of the following relations: a) R = {(2, 7), (8, 3), (5, 5), (4, 3}) and b) R = {(x, x²): x is a prime number less than 15}. Find the domain and range in each of these cases.

Solution:

We know that inverse relation of a relation is obtained by interchanging the first and second elements of the ordered pairs of the given relation. Thus, the inverse of the given relations are,

a) R^-1 = {(7, 2), (3, 8), (5, 5), (3, 4)}.

In this case, domain = {7, 3, 5} and range = {2, 8, 5, 4}.

b) Let us write the given relation in roster form. The list of prime numbers less than 15 are 2, 3, 5, 7, 11, and 13. Thus,

Then R = {(2, 2²), (3, 3²), (5, 5²), (7, 7²), (11, 11²), (13, 13²)} = {(2, 4), (3, 9), (5, 25), (7, 49), (11, 121), (13, 169)}

Now, R^-1 = {(x², x): x is a prime number less than 15} = {(4, 2), (9, 3), (25, 5), (49, 7), (121, 11), (169, 13)}.

Here, domain = {4, 9, 25, 49, 121, 169} and range = {2, 3, 5, 7, 11, 13}

Answer: a) R^-1 = {(7, 2), (3, 8), (5, 5), (3, 4)}, domain = {7, 3, 5} and range = {2, 8, 5, 4}. b) R^-1 ={(4, 2), (9, 3), (25, 5), (49, 7), (169, 13)}, domain = {4, 9, 25, 49, 121, 169} and range = {2, 3, 5, 7, 11, 13}.

Example 2: Find the inverse of the relation R = {(x, y): y = x²}.

Solution:

The relation (R) between x and y is given by the equation y = x².

To find its inverse relation, interchange x and y and solve the resultant equation for y. Then

x = y²

Taking square root on both sides,

±√x = y

Thus, the inverse of the given relation is, R^-1 = {(x, y): y = ±√x}

Answer: R^-1 = {(x, y): y = ±√x}.

Example 3: Find the inverse of the relation that is represented by the following graph.

Solution:

Let us take some points on the graph, say, (0, 1), (2, 4), and (3, 8).

Let us interchange the x and y coordinates to get some points on its inverse.

Then we get (1, 0), (4, 2), and (8, 3).

Plot them on the same graph and join by a curve to get the inverse relation. Also, note that the two curves are symmetric with respect to the line y = x.