Mathematics | 2nd puc | Relations and Functions

 Relations and Functions are important topics in mathematics that involve studying the connections between sets. Here are some examples of relations and functions:

Relations:

Reflexive Relation: Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2)}. Here, R is reflexive because each element of A is related to itself.

Symmetric Relation: Let A = {a, b, c} and R = {(a, b), (b, a), (a, c)}. Here, R is symmetric because if (a, b) is in R, then (b, a) is also in R.

Transitive Relation: Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 3), (1, 3), (3, 4)}. Here, R is transitive because if (1, 2) and (2, 3) are in R, then (1, 3) is also in R.

Functions:

One-to-One Function: Let A = {1, 2, 3} and B = {a, b, c} and f = {(1, a), (2, b), (3, c)}. Here, f is a one-to-one function because each element of A is mapped to a unique element of B.

Onto Function: Let A = {1, 2, 3} and B = {a, b} and f = {(1, a), (2, a), (3, b)}. Here, f is an onto function because every element of B has at least one pre-image in A.

Bijective Function: Let A = {1, 2, 3} and B = {a, b, c} and f = {(1, a), (2, b), (3, c)}. Here, f is a bijective function because it is both one-to-one and onto.

Composition of Functions:

Let f(x) = x^2 and g(x) = x + 1. Then, (f∘g)(x) = f(g(x)) = f(x+1) = (x+1)^2.

Let h(x) = 2x and k(x) = x^3. Then, (k∘h)(x) = k(h(x)) = k(2x) = (2x)^3 = 8x^3.

Inverse Functions:

Let f(x) = 2x + 3. Then, the inverse function of f, denoted by f^(-1), is given by f^(-1)(x) = (x - 3)/2.

Let g(x) = x^3. Then, the inverse function of g, denoted by g^(-1), is given by g^(-1)(x) = cube root of x.

These examples illustrate the different aspects of relations and functions in mathematics. It is important for students to understand these concepts and their applications in various fields.

some problems related to relations and functions. Here are a few examples:

Determine whether the following relations are functions or not:

a) {(1,2), (2,4), (3,6), (4,8)}

b) {(1,2), (2,4), (3,4), (4,6)}

c) {(1,2), (2,4), (3,2), (4,4)}

Solution:

a) This relation is a function because every input (first element in each ordered pair) has exactly one output (second element in each ordered pair).

b) This relation is not a function because the input 3 has two different outputs, 4 and 6.

c) This relation is not a function because the input 1 and 3 have the same output, 2.

Determine the domain and range of the following functions:

a) f(x) = x^2 + 3x - 2

b) g(x) = 1 / (x - 2)

Solution:

a) The domain of f(x) is all real numbers because there are no restrictions on the input values. The range is all real numbers greater than or equal to -3/4 because the minimum value of the function occurs when x = -3/2.

b) The domain of g(x) is all real numbers except 2 because the denominator cannot be zero. The range is all real numbers except 0 because the function cannot output zero.

Find the inverse of the function f(x) = 2x - 1.

Solution:

To find the inverse of a function, we switch the input and output variables and solve for the output variable.

Let y = 2x - 1. Solving for x, we get x = (y + 1) / 2.

Therefore, the inverse of f(x) is f^-1(x) = (x + 1) / 2.

Given the function f(x) = 3x + 2 and 

g(x) = 2x - 1, find the following:

a) (f + g)(x)

b) (f - g)(x)

c) (f * g)(x)

Solution:

a) (f + g)(x) = f(x) + g(x) = 3x + 2 + 2x - 1 = 5x + 1

b) (f - g)(x) = f(x) - g(x) = 3x + 2 - (2x - 1) = x + 3

c) (f * g)(x) = f(x) * g(x) = (3x + 2) * (2x - 1) = 6x^2 + 4x - 3

I hope these examples help you understand relations and functions better. Let me know if you have any further questions!