Class 12 Maths Relations & Functions – NCERT Formulas & Examples
Class 12 Maths Chapter 1: Relations and Functions - Your Ultimate Guide
Struggling with Class 12 Maths Chapter 1? Don’t worry—you’re not alone! At Fuzy Math Academy, we’re here to make Relations and Functions crystal clear with our engaging explanations, step-by-step NCERT solutions, and 24/7 AI Math Solver for Classes 5-12. Let’s dive in and make math fun!
What Are Relations and Functions? The Basics
Think of a relation as a way to pair elements from two sets, like matching students to their favorite subjects. For example, if Set A = {1, 2} and Set B = {a, b}, a relation could be {(1, a), (2, b)}.
A function is a stricter relation: each input gets exactly one output. Imagine assigning each student one unique ID—no duplicates allowed. This chapter lays the foundation for calculus, so let’s get it right!
Key Concepts to Master
Here’s your cheat sheet for Relations and Functions:
- Cartesian Product: All possible pairs from two sets. If A has m elements and B has n, you get m × n pairs.
- Domain: All valid inputs in the relation.
- Range: All outputs produced by those inputs.
- Function Test: Use the vertical line test on graphs—one intersection per x-value.
- Types of Functions: Injective (one-one), surjective (onto), bijective (both). These are key for inverses.
Master these, and you’ll breeze through the chapter.
Essential Formulas to Know
Keep these formulas handy:
- Number of relations from A to B: 2^(m×n).
- Number of functions: n^m.
- Inverse: If f: A → B is bijective, f^{-1}: B → A exists.
- Composition: (f ∘ g)(x) = f(g(x)).
Understand their logic, and you’re set for problem-solving.
Quick Examples to Build Confidence
Let’s try some examples:
Example 1: Is R = {(1, 2), (2, 3), (3, 2)} a function on {1, 2, 3}? Yes—each input has one output. Domain: {1, 2, 3}, Range: {2, 3}.
Example 2: For f(x) = x^2 from reals to non-negative reals. One-one? No, f(2) = f(-2) = 4. Onto? Yes, hits all y ≥ 0.
Example 3: If f(x) = 2x, g(x) = x + 1, then (f ∘ g)(x) = 2(x + 1) = 2x + 2.
25 NCERT Questions with Step-by-Step Solutions
We’ve got you covered with all 25 NCERT questions, from easy to advanced. Try solving first, then click to reveal solutions.
Q1: Determine the domain of R = {(x, x^2) | x ∈ {1, 2, 3}}.
Domain is {1, 2, 3}—the first elements of the pairs.
Q2: Let A = {1, 2}, B = {3, 4}. List A × B.
A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}. Total: 2 × 2 = 4 pairs.
Q3: Is R = {(1, 1), (2, 2), (3, 1)} a function from {1, 2, 3} to {1, 2}?
Yes—each input has one output. Range: {1, 2}.
Q4: Find range of f(x) = 1/x for x ≠ 0.
Domain: ℝ - {0}. Range: ℝ - {0}, as y = 1/x never hits zero.
Q5: Number of relations from {1, 2} to {a, b}.
2^(2 × 2) = 16 subsets of 4 pairs.
Q6: Is f: ℕ → ℕ, f(x) = 2x one-one?
Yes—if f(x) = f(y), 2x = 2y ⇒ x = y. Injective.
Q7: Check if f(x) = x^2 from ℝ to ℝ is onto.
No—negative y (e.g., -1) has no real x since x^2 ≥ 0.
Q8: Find f^{-1} for f(x) = 3x + 4.
y = 3x + 4 ⇒ x = (y - 4)/3. So f^{-1}(x) = (x - 4)/3. Bijective on ℝ.
Q9: (f ∘ g)(x) where f(x) = x - 1, g(x) = x^2.
f(g(x)) = x^2 - 1.
Q10: Domain of f(x) = √(x - 1)/(x - 2).
Square root: x - 1 ≥ 0 ⇒ x ≥ 1. Denominator: x ≠ 2. Domain: [1, 2) ∪ (2, ∞).
Q11: Let f: {1, 2, 3} → {a, b, c}, f(1) = a, f(2) = b, f(3) = c. One-one?
Yes—all outputs distinct, so injective.
Q12: Number of onto functions from 3 elements to 2.
Total functions: 2^3 = 8. Subtract non-onto (constant): 2. So, 8 - 2 = 6.
Q13: Graph y = |x|—function? One-one?
Function: Yes, passes vertical line test. One-one: No, |1| = |-1|. Graph: V-shape at origin.
Q14: If f and g are onto, is f ∘ g onto?
Yes—for any y in codomain, f hits y via some z, and g hits z.
Q15: Find range of R = {(x, y) | y = x + 1, x ∈ ℝ}.
y = x + 1 is a line, range: ℝ.
Q16: Is sin x: ℝ → [-1, 1] bijective?
Onto: Yes, hits all [-1, 1]. One-one: No, periodic (e.g., sin(0) = sin(2π)).
Q17: Number of injections from 3 to 5 elements.
P(5, 3) = 5 × 4 × 3 = 60.
Q18: Domain of cos^{-1}(x).
[-1, 1], required for arccos.
Q19: Check equivalence for "divides" on positive integers.
Reflexive: Yes, x|x. Symmetric: No, 2|4 but not 4|2. Transitive: Yes. Not equivalence.
Q20: (g ∘ f)(x) for f(x) = √x, g(x) = x^2, x ≥ 0.
g(f(x)) = (√x)^2 = x. Identity function.
Q21: Prove if f one-one and g onto, is f ∘ g one-one?
f(g(x)) = f(g(y)) ⇒ g(x) = g(y) (f injective). Doesn’t imply x = y unless g is injective. Counterexample: g(x) = constant (but then not onto unless codomain singleton). Correct: Not necessarily one-one.
Q22: Is R on ℤ: xRy if x + y even, an equivalence relation?
Reflexive: x + x = 2x even. Symmetric: x + y even ⇒ y + x even. Transitive: x + y, y + z even ⇒ x + z even. Yes, equivalence. Classes: evens, odds.
Q23: Modulus function domain/range.
f: ℝ → [0, ∞), domain ℝ, range non-negative reals.
Q24: Number of bijections from 4 to 4 elements.
4! = 24 permutations.
Q25: Show f(x) = e^x is bijection ℝ to (0, ∞).
One-one: Strictly increasing (derivative e^x > 0). Onto: lim x→-∞ e^x = 0+, lim x→∞ = ∞, hits all (0, ∞). Inverse: ln x.
15 FAQs on Relations and Functions
Got questions? We’ve answered the most common ones below:
Q1: What’s the difference between a relation and a function?
A relation pairs elements from two sets; a function ensures each input has exactly one output. Think relation: “friends with many”; function: “one best friend per person.”
Q2: How do I find the domain of real functions?
Check restrictions: denominators ≠ 0, square roots ≥ 0, logarithms > 0. Solve resulting inequalities.
Q3: Why does the vertical line test work?
A vertical line hitting a graph multiple times means one x has multiple y’s—not a function.
Q4: When does a function have an inverse?
It must be bijective (one-one and onto its codomain).
Q5: What does injective mean?
Different inputs produce different outputs—no overlaps.
Q6: How to check if a function is surjective?
Ensure every codomain element is mapped by at least one input.
Q7: Cartesian product for three sets?
A × B × C gives m × n × p triples, like 3D coordinates.
Q8: In composition f ∘ g, does g go first?
Yes—apply g, then f. Read right to left.
Q9: Why are equivalence relations important?
They group elements into classes, like integers modulo n for remainders.
Q10: Is a constant function one-one?
No, unless the domain has one element—all inputs map to one output.
Q11: Range vs. codomain?
Codomain is the declared output set; range is the actual outputs reached.
Q12: When are polynomials onto?
Odd-degree polynomials over reals are onto; even-degree ones may not be if bounded.
Q13: How to prove one-one without graphing?
Show f(x) = f(y) implies x = y using algebra.
Q14: Real-life bijective example?
Assigning unique student IDs to students—one-to-one and onto.
Q15: If f ∘ g = identity, is g ∘ f identity?
Not always—f is a left inverse, so g injective, f surjective. Only identity if both bijective.
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