Relations and Functions Class 11 NCERT Guide

Foundational Definitions: From Sets to Relations

Cartesian Product

For any two non-empty sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. For example, if A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.

Relation

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

Domain

The domain of a relation R from set A to set B is the set of all first elements of the ordered pairs in R. It's a subset of A. It represents all possible 'inputs'.

Codomain

The codomain of a relation R from set A to set B is the entire set B. It represents all possible 'outputs'.

Range

The range of a relation R from set A to set B is the set of all second elements of the ordered pairs in R. It's a subset of the codomain B. It represents the actual 'outputs' you get.

What Makes a Relation a Function?

Now for the most important step: understanding functions. A function is a special kind of relation. Think of a relation as a general connection, but a function is a predictable, reliable connection.

A relation 'f' from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

Let's break this down into two simple rules:

1. Everyone Must Participate: Every element in the domain (set A) must be related to some element in the codomain (set B). No element in A can be left out.
2. No Cheating! An element in the domain (set A) cannot be related to more than one element in the codomain (set B). It can't have two different 'outputs' for the same 'input'.

Imagine a machine (the function) where you put in an object from set A (the input). The machine must produce exactly one object from set B (the output). It can't produce nothing, and it can't produce two different things for the same input. However, it's perfectly fine for two different inputs from set A to produce the same output in set B.

Worked Examples: Identifying Domain, Range, and Functions

• Example 1: Find the domain and range of a relation. Let A = {1, 2, 3, 4, 5} and B = {1, 4, 5}. Let R be the relation 'is less than' from A to B. Find the domain and range of R. Step 1: Write the Cartesian Product A × B and identify the subset for R. The relation R = {(x, y) | x ∈ A, y ∈ B, and x < y}. We list the ordered pairs that satisfy this condition: - For x=1: (1, 4), (1, 5) - For x=2: (2, 4), (2, 5) - For x=3: (3, 4), (3, 5) - For x=4: (4, 5) - For x=5: No y in B is greater than 5. So, R = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}. Step 2: Determine the Domain. The domain is the set of all first elements in the pairs of R. Domain = {1, 2, 3, 4}. Step 3: Determine the Range. The range is the set of all second elements in the pairs of R. Range = {4, 5}. Final Answer: Domain = {1, 2, 3, 4}, Range = {4, 5}.
• Example 2: Check if a relation is a function. Let R = {(1, 3), (2, 5), (3, 7), (4, 9)}. Is this relation a function? If so, what is its domain and range? Step 1: Check the first condition for a function. Every element in the set of first components {1, 2, 3, 4} must have an image. Here, 1 is mapped to 3, 2 to 5, 3 to 7, and 4 to 9. No first element is repeated. Step 2: Check the second condition for a function. Each first element must have only one image. - 1 maps only to 3. - 2 maps only to 5. - 3 maps only to 7. - 4 maps only to 9. Since each input has exactly one output, the relation is a function. Step 3: Find the domain and range. Domain = The set of first elements = {1, 2, 3, 4}. Range = The set of second elements = {3, 5, 7, 9}. Final Answer: Yes, it is a function. Domain = {1, 2, 3, 4}, Range = {3, 5, 7, 9}.
• Example 3: Check if a relation is NOT a function. Let R = {(1, a), (1, b), (2, c)}. Is this relation a function? Step 1: Check the conditions for a function. We look at the first elements of the ordered pairs. The element '1' from the domain is associated with two different elements in the codomain, 'a' and 'b'. Step 2: Apply the function definition. A function requires that every element of the domain has one and only one image. Since the input '1' has two outputs ('a' and 'b'), this relation violates the definition of a function. Final Answer: No, this relation is not a function because the element 1 has two distinct images.

Exam Traps and Common Mistakes

1. Confusing Range and Codomain: Remember, the codomain is the entire set B of possible outputs. The range is the subset of B that are actual outputs. In an exam question, if set B = {1, 2, 3, 4, 5} and your relation only produces outputs {1, 3, 5}, then the range is {1, 3, 5}, not the whole set B.

2. The 'Every Input' Rule: Students often check if one input has multiple outputs, but forget to check if every possible input has at least one output. For a relation f from A to B to be a function, every single element in set A must appear as a first element in one of the ordered pairs.

3. The Vertical Line Test: For functions plotted on a graph, use the Vertical Line Test. If you can draw a vertical line anywhere on the graph that intersects the curve at more than one point, it is NOT a function. This is a quick visual check for the 'one input, one output' rule.

Practice Questions with Solutions

• Q: Let A = {1, 2, 3} and B = {a, b}. How many relations can be defined from A to B? A: Step 1: Find the number of elements in each set. n(A) = 3 and n(B) = 2. Step 2: Calculate the number of elements in the Cartesian product A × B. The number of elements in A × B is n(A) × n(B) = 3 × 2 = 6. Step 3: Understand that a relation is a subset of A × B. The number of possible subsets of a set with 'k' elements is 2^k. Step 4: Apply this to A × B. The number of possible relations is the number of possible subsets of A × B, which is 2^6. Final answer: 2^6 = 64 relations can be defined from A to B.
• Q: Examine the relation R = {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}. Is it a function? State the domain and range. A: Step 1: Check if the relation is a function. We need to see if each first element has one and only one image. The first elements are 2, 5, 8, 11, 14, 17. None of these are repeated. Each distinct input has exactly one output (which happens to be '1' in all cases). This is allowed. Step 2: Conclude if it's a function. Since both conditions are met (every input is used, and each has only one output), the relation R is a function. This is an example of a constant function. Step 3: Find the domain. The domain is the set of all first elements: {2, 5, 8, 11, 14, 17}. Step 4: Find the range. The range is the set of all second elements. Since the only second element is 1, the range is {1}. Final answer: Yes, it is a function. Domain = {2, 5, 8, 11, 14, 17}, Range = {1}.
• Q: Let f(x) = x² and g(x) = 2x + 1 be two real functions. Find (f + g)(x) and (fg)(x). A: Step 1: Define the sum of two functions. (f + g)(x) is defined as f(x) + g(x). Step 2: Substitute the given functions. (f + g)(x) = x² + (2x + 1). Step 3: Simplify the expression. (f + g)(x) = x² + 2x + 1. This can also be written as (x+1)². Step 4: Define the product of two functions. (fg)(x) is defined as f(x) g(x). Step 5: Substitute the given functions. (fg)(x) = (x²) (2x + 1). Step 6: Simplify the expression by distributing. (fg)(x) = 2x³ + x². Final answer: (f + g)(x) = x² + 2x + 1 and (fg)(x) = 2x³ + x².
• Q: Find the domain of the real function f(x) = √(9 - x²). A: Step 1: Understand the constraint for a square root function. The expression inside a square root must be non-negative (greater than or equal to zero) for the function to have a real value. Step 2: Set up the inequality. So, we must have 9 - x² ≥ 0. Step 3: Solve the inequality. 9 ≥ x² This is equivalent to x² ≤ 9. Taking the square root of both sides, we get |x| ≤ 3. Step 4: Interpret the absolute value inequality. |x| ≤ 3 means that x must be between -3 and 3, inclusive. So, -3 ≤ x ≤ 3. Step 5: Write the domain in interval notation. The domain is [-3, 3]. Final answer: The domain of the function is the interval [-3, 3].

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