CBSE Class 12 · Mathematics
Relations and Functions
Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.
This lesson follows the official chapter “Relations and Functions” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 12
- Medium level
- 8 concepts
1Types of Relations
Types of relations on a set: empty, universal, reflexive, symmetric, transitive, equivalence as taught. A type is a test on pairs, not a mood. On {1,2,3}, {(1,1),(2,2),(3,3)} is reflexive; add (1,2) and (2,1) and you have started symmetric. Equivalence needs all three.
Calling any list of pairs “equivalence” without the three tests is a miss.
Figure. A relation on {1,2,3} is the filled cells of a pair table. The diagonal is the identity pairs that a reflexive test needs. Cells (1,2) and (2,1) together are one symmetric pair. No self-loop is drawn.
How it works
- Name the test (reflexive / symmetric / transitive)The type.
- Run it on the pair-listIn or out.
- Keep equivalence as all threeThe bundle.
2Types of Functions
Types of functions: one-one (injective), onto (surjective), bijection as taught. f:{1,2,3}→{a,b,c} with f(1)=a, f(2)=b, f(3)=c is both. A type is a pair-count test, not a graph-pretty.
A vertical-line pass as “one-one” is the wrong line-test.
Figure. A function is one-one when distinct inputs keep distinct outputs, and onto when every codomain point is hit. The left map a,b,c → 1,2,3 is both. The right map sends a and b to the same 1, so it is many-one.
How it works
- Ask distinct inputs → distinct outputsOne-one.
- Ask every output is hitOnto.
- Both → bijection as taughtThe type.
3Composition of Functions and Invertible Function
Composition: (g∘f)(x)=g(f(x)) as taught. Invertible means a two-sided inverse exists — bijection on these school sets. If f(x)=x+3 on reals, f⁻¹(x)=x−3. Composition is apply-then-apply, not a product of values.
Writing g∘f as g×f is a miss.
Figure. Composition is the one jump g ∘ f along A → B → C. Invertible is a two-way pair: f and f⁻¹ between the same two sets, so each undoes the other. Both tests sit on maps, not on a formula name.
How it works
- Apply f first, then gComposition.
- Invert only if one-one and onto as taughtInvertible.
- Check f⁻¹(f(x))=xThe inverse-test.
Inverse of x+3
f(x)=x+3 on reals. Find f⁻¹(8).
- f(x)=x+3y=x+3
- Solve x=y−3f⁻¹(y)=y−3
- f⁻¹(8)5
Pro tip. Undo add-3 by subtract-3.
4A definition is a test you can run
Relation / function-word is a test: a pair-list type, a one-one/onto, or a compose-inverse. If you only say “arrow”, you have a heading.
A mapping-sticker is not the test.
Figure. “Function” is a test you can run on a pairing: each domain point has exactly one arrow. Two arrows from the same input fail the test, even if someone wrote the word function on the page.
How it works
- Name the type or the composeThe object.
- Give the school sentenceThe test.
- Then the word has contentThe definition ran.
5Name the given before the unknown
The given is a pair-list or a rule. The unknown is the type or the inverse. Copy {(1,2),(2,1)} before you treat it as reflexive.
Using f(x)=x² as onto on reals without checking negatives is a silent miss.
Figure. Name the given before the unknown: write the two sets and the pairing rule first. Range and onto are questions you ask after the given is on the page, not names you invent first.
How it works
- Copy the pairs or the ruleThe given.
- Name type or inverseThe unknown.
- Then testGiven first.
6One worked case is enough at this class
One worked case is enough: f(x)=2x+1, f⁻¹(x)=(x−1)/2. Do not stack four more inverses. The case teaches undo-the-rule.
A poster of ten arrows is not more science.
Figure. One worked pairing is enough at this class: R = {(1,2),(2,3)} already lets you run the tests. A pile of extra cases is not a new idea.
How it works
- Take one bijectionThe case.
- Write the inverse by undoingThe teach.
- Stop — one caseThis class.
7A miss: swapping the school name for the picture
A miss: swapping the school name for a picture. “Reflexive” is every (a,a) present — a pretty loop-drawing is a setting. Keep the name the test can run.
A Venn-glow without pair-talk is a mute picture.
Figure. A miss is swapping the school name for the picture: writing “function” while the pairing shows two outputs from one input. Believe the arrows, not the title.
How it works
- Keep the taught name (reflexive, one-one, …)The science.
- Use a picture only as a settingHonest.
- Refuse a swap of name for glowThe miss named.
8Check by the opposite action or the opposite test
Check by the opposite: not reflexive if some (a,a) is missing; not one-one if two inputs share an output. The opposite test is the disproof.
A single lucky pair as “proved equivalence” is a miss.
Figure. Check an inverse by the opposite map: apply f, then f⁻¹, and you must land on the same x. If the return trip misses, the pair is not inverse.
How it works
- Name the type you claimThe claim.
- Look for a counter-pairThe opposite.
- If a counter sits, the type failsThe check.
f(x)=x+3 on reals has inverse
- x−3
- x+3
- 3x
Undo add-3.
Notes
- The official chapter title is “Relations and Functions”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- (g∘f)(x)=g(f(x))
- f⁻¹(f(x))=x when invertible
Recap
Hold these pegs from the official chapter “Relations and Functions”. The wording is ExamMaster’s teaching, not a textbook recap.
- Types of Relations
- Types of relations on a set: empty, universal, reflexive, symmetric, transitive, equivalence as taught.
- Types of Functions
- Types of functions: one-one (injective), onto (surjective), bijection as taught.
- Composition of Functions and Invertible Function
- Composition: (g∘f)(x)=g(f(x)) as taught. Invertible means a two-sided inverse exists — bijection on these school sets. If f(x)=x+3 on reals, f⁻¹(x)=x−3. Composition is apply-then-apply, not a product of values.
- A definition is a test you can run
- Relation / function-word is a test: a pair-list type, a one-one/onto, or a compose-inverse.
- Name the given before the unknown
- The given is a pair-list or a rule. The unknown is the type or the inverse. Copy {(1,2),(2,1)} before you treat it as reflexive.
- One worked case is enough at this class
- One worked case is enough: f(x)=2x+1, f⁻¹(x)=(x−1)/2.
Practise Relations and Functions
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device