Chapter 2 — Functions Overview
Subject
Mathematics (ISC)
Chapter
2 — Functions
Textbook
Chand High School
Dates
23–27 Mar 2026
Learning Objectives
1. Recall types of relations and function concepts from Chapter 1 and Class XI.
2. Distinguish one-one (injective), onto (surjective) and bijective functions with examples.
3. Count one-one, onto and bijective functions using formulas (ⁿPₘ, m!, inclusion-exclusion).
4. Prove a given function is one-one / onto / bijective using standard algebraic method.
5. Define and compute composite functions (f∘g, g∘f); verify associativity and non-commutativity.
6. Define invertible functions; establish necessary & sufficient condition — bijection.
7. Find the inverse of a given function and verify using composition; identify self-inverse functions.
8. Solve application problems including domain restriction questions.
9. Attempt competency-based, Assertion–Reason, Case Study and MCQ-style questions.
2. Distinguish one-one (injective), onto (surjective) and bijective functions with examples.
3. Count one-one, onto and bijective functions using formulas (ⁿPₘ, m!, inclusion-exclusion).
4. Prove a given function is one-one / onto / bijective using standard algebraic method.
5. Define and compute composite functions (f∘g, g∘f); verify associativity and non-commutativity.
6. Define invertible functions; establish necessary & sufficient condition — bijection.
7. Find the inverse of a given function and verify using composition; identify self-inverse functions.
8. Solve application problems including domain restriction questions.
9. Attempt competency-based, Assertion–Reason, Case Study and MCQ-style questions.
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⚠Holidays: 28 Mar (Eid) and 1 Apr (Ram Navami) — no classes on these days.
DAY 1 · FRI 27 MAR 2026
Day 1 — One-one · Onto · Bijective
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One-one · Onto · Bijective — Key Definitions
One-onef(x₁)=f(x₂) ⟹ x₁=x₂; equivalently x₁≠x₂ ⟹ f(x₁)≠f(x₂)
OntoRange of f = Co-domain; every y∈B has a pre-image in A
Bijectivef is both one-one and onto
Count 1-1ⁿ⁽ᴮ⁾Pn(A) when n(A)≤n(B); 0 otherwise
Count Bijn! when n(A)=n(B)=n; 0 otherwise
Graph TestOne-one ⟺ any horizontal line meets graph at most once
Day 1 — Examples
Ex 1
Prove f : ℝ→ℝ, f(x) = 2x + 6 is one-one. [Graphical — horizontal line test]
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Ex 2
Show f : ℤ→ℤ, f(x) = x² + x is many-one.
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Ex 3(A)
Number of one-one functions: (i) n(A)=3, n(B)=3 (ii) n(A)=3, n(B)=4 (iii) n(A)=4, n(B)=3.
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Ex 5
State whether f : ℕ→ℕ, f(x) = 5x is injective or surjective.
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Ex 6
Check injectivity and surjectivity: (i) f:ℕ→ℕ, f(x)=x² (iii) f:ℝ→ℝ, f(x)=x² (iv) f:ℕ→ℕ, f(x)=x³.
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Ex 7
Show f:ℝ→ℝ, f(x) = 2x/(1+x²) is neither one-one nor onto. Find set A so that f:ℝ→A is onto.
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Ex 10
f:ℝ→ℝ, f(x) = 4x³+7; show f is a bijection.
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Ex 11
State whether f:ℝ→ℝ, f(x) = 1+x² is one-one / onto / bijective.
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Ex 13
Show f:ℕ→ℕ, f(x) = x+1 (x odd), x−1 (x even) is bijective.
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Ex 19
Number of bijective functions from A={a,b,c} to B={x,y,z}; and when n(B)=4.
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Day 1 — Exercise 2(A)
Q 2
★ f:A→B, A={1,2,3,4}, B={1,2,3,4,5,6}, f(x)=x. Identify the type of function.
(a) One-one but not onto
(b) Onto but not one-one
(c) Both one-one and onto
(d) Neither one-one nor onto
Q 6
f:ℝ→ℝ, f(x)=(2x−7)/4; show f is one-one and onto.
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Q 9
A=ℝ−{3}, B=ℝ−{1}, f:A→B, f(x)=(x−2)/(x−3); is f one-one and onto?
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Q 17
★ A has 6 distinct elements; number of distinct functions that are NOT bijective = ?
(a) 6!
(b) 6⁶−6!
(c) 6⁶
(d) 6!−6
Q 18
A={1,3,5,7}, B={1,2,...,8}; find the number of one-to-one functions from A to B.
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Q 19
★ A={x∈ℕ:x≤5}, B={x∈ℤ:x²−5x+6=0}; number of onto functions from A to B = ?
(a) 2⁵−2
(b) 2⁵
(c) 30
(d) 0
Q 20
★ x = no. of one-one functions from A(3 elements) to B(5 elements); y = one-one from A to A×B. Find relation between x and y.
(a) y = 10x
(b) y = x
(c) 2y = x
(d) y = 2x
Day 1 — Homework
HW 1
Ex 2(A) Q1(a)–(b): For 'is greater than' and 'is the square of' — state R / S / T / E / N.HW 2
Ex 2(A) Q4: f:ℝ→ℝ, f(x)=3x+4; show f is bijective and find f⁻¹.HW 3
Revise counting formulas: one-one (ⁿPₘ), onto (inclusion-exclusion) and bijections (n!).Day 1 — Video Resources
Day 1 Exit Ticket
1Define one-one and onto in your own words.
2How many bijections from {a,b,c} to {x,y,z}?
3Is f(x)=x² one-one on ℝ? Why?
DAY 2 · MON 30 MAR 2026
Day 2 — Composition of Functions
Concept Board — Day 2
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Composition of Functions — Key Properties
g∘f(g∘f)(x) = g(f(x)) — apply f first, then g
Domaindom(g∘f) = {x∈dom(f) : f(x)∈dom(g)}
Non-comm.In general f∘g ≠ g∘f
Assoc.h∘(g∘f) = (h∘g)∘f — always holds
PropertyIf f, g both one-one (onto) ⟹ g∘f also one-one (onto)
Day 2 — Examples: Composition of Functions
Ex 20
f={(1,2),(3,5),(4,1)}, g={(1,3),(2,3),(5,1)}; write g∘f as a set of ordered pairs.
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Ex 21(i)
f(x)=|x|, g(x)=|5x−2|; find g∘f and f∘g.
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Ex 21(ii)
f(x)=8x³, g(x)=x^(1/3); find g∘f and f∘g.
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Ex 23
f(x)=x²−1, g(x)=√x; find (a) f∘g (b) g∘f (c) f∘f (d) g∘g.
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Ex 24
f(x)=4x²+1, g(x)=1/(x+2); find f∘g(x) and g∘f(x).
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Ex 25
f(x)=sin x, g(x)=x²; prove g∘f ≠ f∘g.
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Ex 28
★ f(x)=log[(1+x)/(1−x)], g(x)=(3x+x³)/(1+3x²); find f{g(x)} = ?
(a) f(x)
(b) 2f(x)
(c) 3f(x)
(d) [f(x)]³
Ex 30
★ f(x)=(x+1)/(x−1), x≠1; (f∘f∘f∘f)(x) = ?
(a) x
(b) 1/x
(c) (x+1)/(x−1)
(d) (x−1)/(x+1)
Ex 31
f:x→2x, g:x→x², h:x→x+1; verify h∘(g∘f) = (h∘g)∘f. [Associativity]
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Ex 32
f(x)=2x+3, g(x)=x²+7; find x for which f(g(x)) = 25.
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Ex 33
f=GIF (greatest integer function), g=|x|; find (f∘g)(−3/2)+(g∘f)(4/3).
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Day 2 — Exercise 2(B)
Q 1
f:ℕ→ℝ, f(x)=(2x−1)/2; g:ℚ→ℝ, g(x)=x+2; find (f∘g)(3/2).
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Q 3(i)
f(x)=3x−2, g(x)=x²; find g∘f(3), f∘g(1) and f∘f(0).
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Q 4
f(x)=x+5, g(x)=x²−3; find f(g(0)), g(f(0)), f(g(x)) and g(f(x)).
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Q 7
★ f(x)=(2x+1)/(3x−2); find (f∘f)(2) = ?
(a) 2
(b) 1
(c) 3/2
(d) 0
Q 8
★ f(x)=(1−x)/(1+x); f{f(cos 2θ)} = ?
(a) cos 2θ
(b) tan²θ
(c) cot²θ
(d) sin 2θ
Q 10
★ g(x)=x²+1, f(x)=x−3; find x for which g(f(x)) = 10.
(a) x=±1
(b) x=6 or x=0
(c) x=3±√9
(d) x=±2
Q 11
★ f(x)=sin²x; g[f(x)]=|sin x|; g(x) = ?
(a) √x
(b) x²
(c) |x|
(d) x
Q 14
f(x)=(3x+4)/(5x−7), g(x)=(7x+4)/(5x−3); show g∘f = f∘g.
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Q 16
f(x)=x³, g(x)=2x²+1; find f∘g(x) and g∘f(x).
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Day 2 — Homework
HW 1
Q12: f(x)=x²+3x+1, g(x)=2x−3; find f∘g and g∘f.HW 2
Q15: f(x)=x²+2, g(x)=3x−1; show f∘g ≠ g∘f.HW 3
Memorise the rule: g∘f means apply f first, then g.Day 2 — Video Resources
Day 2 Exit Ticket
1What does g∘f mean? Which is applied first?
2Is f∘g always equal to g∘f? Give example.
3Find (f∘g)(0) if f(x)=2x, g(x)=x²+1.
DAY 3 · TUE 31 MAR 2026
Day 3 — Invertible Functions
Concept Board — Day 3
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Invertible Functions — Key Theory
Invertiblef:A→B is invertible iff it is bijective
f⁻¹f⁻¹:B→A satisfies f⁻¹(f(x))=x ∀x∈A and f(f⁻¹(y))=y ∀y∈B
Verifyf and g are inverses ⟺ f∘g=I_B and g∘f=I_A
Self-inv.f∘f=I_A ⟹ f=f⁻¹
GraphGraph of f⁻¹ is reflection of graph of f in y=x
4 Steps(1) Write y=f(x) (2) Solve for x (3) Replace y by x (4) State domain & range
Day 3 — Examples: Invertible Functions
Ex 36
Using arrow diagrams, state whether the following have inverses: (i) f:{1,2,3,4}→{10}, f={(1,10),(2,10),(3,10),(4,10)} (ii) g:{5,6,7,8}→{1,2,3,4}, g={(5,4),(6,3),(7,4),(8,2)} (iii) h:{2,3,4,5}→{7,9,11,13}, h={(2,7),(3,9),(4,11),(5,13)}.
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Ex 39
f(x)=2x/(3−x), g(x)=3x/(x+2); use composition to verify f and g are inverses.
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Ex 42
f(x)=(4x+3)/(6x−4), x≠2/3; show f∘f(x)=x. What is the inverse of f?
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Ex 43
f(x)=(3x−4)/5 is invertible; write f⁻¹(x).
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Ex 44
f(x)=√(2x−3); find f⁻¹(x) and its domain.
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Ex 45
★ f(x)=(x+1)², x≥−1; g(x) is reflection of f in y=x; g(x) = ?
(a) (x+1)²
(b) √x−1
(c) √x+1
(d) (x−1)²
Ex 49
f:ℝ→ℝ, f(x)=4x−7; show f is invertible and find f⁻¹.
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Ex 50
A=ℝ−{3}, B=ℝ−{1}, f(x)=(x−2)/(x−3); show one-one and onto; find f⁻¹.
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Ex 51
f:ℝ₊→[−5,∞), f(x)=9x²+6x−5; show f invertible; find f⁻¹(y).
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Day 3 — Exercise 2(C)
Q 1(i)
f:ℝ→ℝ, f(x)=2x+3; find f⁻¹(x).
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Q 1(iii)
f:ℝ→ℝ, f(x)=(3x+5)/2; find f⁻¹(x).
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Q 3
f:ℝ→ℝ is invertible, f(x)=(1−x)^(1/3); find f⁻¹(x).
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Q 4
f(x)=[4−(x−7)³]^(1/5); find f⁻¹(x).
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Q 7
★ f:ℝ→ℝ, f(x)=|x|; then f⁻¹(x) = ?
(a) |x|
(b) ±x
(c) f⁻¹ does not exist
(d) x
Q 8
f:ℝ→ℝ, f(x)=4x+3; show f is invertible and find f⁻¹.
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Q 10
A=ℝ−{2}, B=ℝ−{1}, f(x)=(x−1)/(x−2); show f is onto; find f⁻¹.
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Q 11
A=ℝ−{2/3}, f(x)=(4x+3)/(6x−4); show one-one and onto; find f⁻¹.
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Q 14
Show that the inverse of f(x)=(2x+1)/(3x−2) is f itself. [Self-inverse]
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Q 15(i)
f(x)=2x−6, g(x)=x/2+3; use composition to show f and g are inverses.
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Day 3 — Homework
HW 1
Q6: f:ℝ→ℝ, f(x)=x³; find f⁻¹(8).HW 2
Remember: f is invertible if and only if it is bijective. Prove or disprove in each case.HW 3
Practice the 4-step method for finding f⁻¹ on all textbook exercises.Day 3 — Video Resources
Day 3 Exit Ticket
1What condition makes f invertible?
2Write the 4 steps to find f⁻¹.
3If f∘f=I, what can you say about f?
DAY 4 · THU 2 APR 2026
Day 4 — Chapter Review & Competency
Concept Board — Day 4
Click to open · Draw diagrams & explain concepts
▶
Day 4 — Chapter Review: VSA / Fill in the Blank
VSA Q1(i)
Only __________ functions are invertible.
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VSA Q1(ii)
f(x)=x+3, g(x)=x−3; f∘g(3) = ___.
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VSA Q1(iii)
f:ℝ→ℝ, f(x)=2x+3; f⁻¹(x) = ___.
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VSA Q1(x)
Number of one-one functions from A={a,b,c} to B={x,y,z} = ___.
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VSA Q1(xii)
Number of bijections from A={a,b,c} to B={x,y,z} = ___.
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Day 4 — True or False
Q 2
Every function is invertible. [True / False — justify]
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Q 3
f:ℝ→ℝ, f(x)=sin x is bijective and invertible. [True / False — justify]
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Q 5
If f and g are two bijections such that (g∘f) exists, then (g∘f)⁻¹ = f⁻¹∘g⁻¹. [True / False]
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Q 7
f(x)=27x³ and g(x)=x^(1/3); (g∘f)(x)=x. [True / False — justify]
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Day 4 — Review Examples
Ch. Review Q8
f(x)=x³+1; find f⁻¹(−215), f⁻¹(65) and f⁻¹(10.37).
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Ch. Review Q11
f:ℝ−{3}→ℝ−{1}, f(x)=(x−2)/(x−3); g:ℝ→ℝ, g(x)=2x−3; find all x for which f⁻¹(x)+g⁻¹(x) = 13/2.
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Ex 2A Q13
f:ℝ→ℝ, f(x)=2x³−5; show f is a bijection.
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Ex 2B Q12
f(x)=x²+3x+1, g(x)=2x−3; find f∘g(x) and g∘f(x).
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Ex 2C Q8
f:ℝ→ℝ, f(x)=4x+3; show f is invertible and find f⁻¹.
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Day 4 — Assertion–Reason Questions
AR (i)
Assertion: Every bijective function has an inverse. Reason: A function has an inverse iff it is bijective. [State A/R correct or incorrect and give the correct reason]
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AR (iii)
Assertion: f:ℝ₊→[−5,∞), f(x)=5x²+6x−9 is invertible with f⁻¹(y)=(√(54+5y)−3)/5. Reason: A function is invertible iff it is bijective.
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AR (iv)
Assertion: f(x)=2x+3, g(x)=x²+7; f(g(x))=25 ⟹ x=±1. Reason: Inverse of a bijection is also a bijection.
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Day 4 — Case Study Q14: Inter-school Mathematics Quiz
Context
Team A: 5 boys — b₁,b₂,b₃,b₄,b₅. Team B: 4 girls — g₁,g₂,g₃,g₄. Pairing function f = {(b₁,g₁),(b₂,g₃),(b₃,g₂),(b₄,g₄),(b₅,g₄)}.
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Q14(i)
★ The mapping f is:
(a) one-one not onto
(b) only onto
(c) both one-one and onto
(d) neither one-one nor onto
Q14(ii)
★ Number of functions from A to B:
(a) 20
(b) 5⁴
(c) 4⁵
(d) ⁵P₄
Q14(iii)
★ Number of functions from B to A:
(a) 20
(b) 5⁴
(c) 4⁵
(d) ⁵P₄
Q14(iv)
★ Number of one-one functions from A to B:
(a) 4⁵
(b) ⁵P₄
(c) 0
(d) 5⁴
Day 4 — Case Study Q15: School Prefects
Context
M = {Rahul, Piyush, Samir}, F = {Shweta, Arushi}. Answer Q15(i)–(v): total relations, total functions, type of given function, injective count, bijective count.
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Q15(i)
★ Total number of relations from M to F:
(a) 5
(b) 6
(c) 2⁶
(d) 2⁵
Q15(ii)
★ Total number of functions from M to F:
(a) 6
(b) 2³
(c) 3²
(d) 2⁶
Q15(iii)
★ Is f(x) = Shweta for all x ∈ M a function? If yes, what type?
(a) Bijective
(b) One-one not onto
(c) Onto not one-one
(d) Neither one-one nor onto
Q15(iv)
★ Number of injective (one-one) functions from M to F:
(a) 0
(b) 2
(c) 6
(d) 3
Q15(v)
★ Number of bijective functions from M to F:
(a) 0
(b) 6
(c) 2
(d) 3!
Day 4 — Homework & Self-Study
HW 1
Complete all pending Exercise 2(A), 2(B) and 2(C) questions.HW 2
Attempt Assertion–Reason questions AR(i) to AR(vi) from the chapter review.HW 3
Read Case Study Q14 and Q15 and answer all parts with full justification.HW 4
Prepare for Chapter 2 test covering all topics — Days 1–4.Day 4 — Video Resources
Day 4 Exit Ticket
1State the condition for a bijective function.
2True/False: sin x is bijective on ℝ.
3Find f⁻¹ if f(x)=3x−4.