Relations and Functions — Lesson
1) Hook — A Fun Real-Life Example
Imagine you are organizing a cricket tournament in your school. Each player is assigned a unique jersey number, and each jersey number corresponds to exactly one player. Here, the relationship between jersey numbers and players is a function because every jersey number maps to one player only. But if you consider the relationship between players and their favorite cricket shots, a player might like multiple shots, so this relation is not a function.
2) Core Concepts — Relations and Functions Explained
Relation: A relation from set A to set B is a subset of the Cartesian product A × B. It associates elements of A with elements of B.
Example: Let A = {1, 2, 3} be students and B = {A, B, C} be their grades. A relation R could be {(1, A), (2, B), (3, A)}.
| Student (A) | Grade (B) |
|---|---|
| 1 | A |
| 2 | B |
| 3 | A |
Function: A function is a special type of relation where every element of the domain (set A) is related to exactly one element of the codomain (set B).
Example: Consider the function f: ℝ → ℝ defined by f(x) = 2x + 3. For every real number x, there is exactly one output.
| x | f(x) = 2x + 3 |
|---|---|
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
Types of Functions:
- One-to-One (Injective): Different inputs have different outputs.
- Onto (Surjective): Every element of codomain has a pre-image.
- Bijective: Both one-to-one and onto.
Domain, Codomain, Range:
- Domain: Set of all possible inputs.
- Codomain: Set in which outputs lie.
- Range: Actual set of outputs produced.
3) Key Formulas / Rules
Function Definition: f: A → B is a function if ∀a ∈ A, ∃! b ∈ B such that (a, b) ∈ f.
One-to-One (Injective): f(a₁) = f(a₂) ⇒ a₁ = a₂
Onto (Surjective): ∀b ∈ B, ∃a ∈ A such that f(a) = b
Bijective: Function is both injective and surjective.
Composite Function: If f: A → B and g: B → C, then (g ∘ f)(x) = g(f(x))
Inverse Function: If f is bijective, then inverse f⁻¹: B → A exists such that f⁻¹(f(x)) = x
4) Did You Know?
Functions are everywhere in Indian culture! The ancient Indian mathematician Brahmagupta (7th century) studied functions related to astronomy, like the sine function, which helped in calculating planetary positions — a key part of the Indian calendar system.
5) Exam Tips — Common Mistakes and Board Patterns
- Common Mistake: Confusing relation with function. Remember, every input must have exactly one output for a function.
- Check Domain and Range: Always specify domain and range clearly in function questions.
- Inverse Function: Verify if the function is bijective before finding the inverse.
- Composite Functions: Pay attention to the order: (g ∘ f)(x) = g(f(x)), not f(g(x)).
- Board Exam Pattern: Questions include:
- Definition and examples of relations and functions.
- Finding domain, range, and inverse of functions.
- Proving injectivity or surjectivity.
- Composite functions and their evaluation.
- Previous Year Question Tip: CBSE often asks to verify if a given relation is a function or not, and to find inverse functions for linear and quadratic functions.
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