Functions

• The precise definition of a function f: A → B as a relation where every input has exactly one output.
• To classify functions as one-one (injective), onto (surjective), and bijective — core JEE vocabulary.
• To find domain and range of real functions involving fractions, square roots, and modulus.
• To compose functions (fog, gof) and determine inverse functions when they exist.

Level 1 — Function vs relation

Verbal: A function from A to B assigns to each element of A exactly one element of B. If any input maps to two outputs, it is only a relation, not a function.

Symbolic: f: A → B means f(a) = b is unique for each a ∈ A. The set A is the domain; B is the co-domain; the range is {f(a) : a ∈ A} ⊆ B.

Vertical line test (graphs): A graph in the xy-plane represents y as a function of x if every vertical line meets the graph at most once.

Examples:

• f(x) = x² is a function ℝ → ℝ.
• x = y² (parabola opening right) is not a function of x (e.g., x = 4 gives y = ±2).

Level 2 — Types and operations

Composition: (fog)(x) = f(g(x)). Domain of fog: x must be in domain of g, and g(x) in domain of f.

Inverse: f⁻¹ exists iff f is bijective. (f⁻¹of)(x) = x.

Domain rules (real functions):

• Denominator ≠ 0
• Even root: radicand ≥ 0
• log: argument > 0

NCERT standard: f(x) = 1/(x−2) → domain ℝ − {2}. f(x) = √(3−x) → domain (−∞, 3].

NCERT spotlight — Domain, range, and piecewise functions

The modulus function f(x) = |x| has range [0, infinity) and is not one-one on all real numbers. The greatest integer function [x] is piecewise constant with domain R and range Z.

Rational function domain: For f(x) = (x+1)/(x-2), exclude x = 2. For f(x) = sqrt(x-3), require x >= 3. When composing fog, x must lie in the domain of g and g(x) must lie in the domain of f.

Graph tests: A function is one-one if every horizontal line meets its graph at most once. If f: A to B is bijective, then |A| = |B|.

Let f: ℝ → ℝ, f(x) = 2x + 1 and g: ℝ → ℝ, g(x) = x². Find (fog)(3) and (gof)(3), and state whether f is one-one and onto.

Step 1 — (fog)(3) = f(g(3)) = f(9) = 2(9) + 1 = 19.
Step 2 — (gof)(3) = g(f(3)) = g(7) = 49.
         Note: fog ≠ gof in general (not commutative).
Step 3 — One-one: f(a) = f(b) ⇒ 2a+1 = 2b+1 ⇒ a = b → injective ✓
Step 4 — Onto: for y ∈ ℝ, solve y = 2x+1 → x = (y−1)/2 ∈ ℝ → surjective ✓
         f is bijective; f⁻¹(y) = (y−1)/2.

Applications — cost and revenue models

If C(x) = 50 + 3x is cost and R(x) = 5x is revenue for x items, profit P(x) = R(x) - C(x) = 2x - 50. Domain of x is non-negative integers in business context. Break-even when P(x) = 0 gives x = 25. Function composition models nested processes: temperature conversion then scaling in scientific formulas.

Review and practice drill

Review checklist: (1) Domain from restrictions: denominators, even roots, log arguments. (2) Onto requires solving y = f(x) for x in domain for every y in codomain. (3) Bijection gives inverse function f inverse. (4) Composition order: (fog)(x) means apply g first. Practice: f(x) = 1/(x-3), g(x) = sqrt(x). Domain of g is x>=0; fog requires x>=0 and g(x) not equal 3 — exclude if sqrt(x)=3 so x not equal 9 when combined with domain.

• Is the relation {(1,2), (2,3), (1,4)} from {1,2} to {2,3,4} a function? Why?
• Find domain of f(x) = √(4 − x²) + 1/(x − 1).
• If f = {(1,a), (2,b), (3,a)} is a function from {1,2,3} to {a,b}, is f onto {a,b}?

Open the Practice tab for graded questions on Functions.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.