Relations and Functions: Class 11 Maths Chapter 2 Hinglish mein

Mathematics · Hinglish notes, simulations aur practice · Official NCERT chapter PDF ↗

Published 18 Oct 2026 · Updated 18 Oct 2026

85 min
1 simulation
42 practice questions
21 MCQs

Is chapter mein aap kya seekhenge

Do ya teen finite sets ka Cartesian product banao, uske elements gino aur ordered pairs ki equality ka use karo.
Relation ko A × B ke subset ki tarah roster form, set-builder form aur arrow diagram mein likho, aur uska domain, codomain aur range nikalo.
Ek finite set se doosre finite set tak ke relations gino.
Decide karo ki koi relation function hai ya nahi, aur real function ka domain aur range nikalo.
Identity, constant, polynomial, rational, modulus, signum aur greatest integer functions ke graphs banao aur padho.
Real functions ko add, subtract, multiply aur divide karo aur har result ka domain batao.
Values se linear function nikalo aur tukdon mein define kiye gaye functions ka graph banao.

NCERT Mathematics Class 11, Chapter 2, pp. 24–42 par based original companion text. Yeh NCERT se juda nahi hai aur NCERT ne ise endorse nahi kiya hai.

2.1 Introduction

Railway reservation chart har berth ko ek passenger se milata hai, marks register har roll number ko ek score se milata hai, aur tumhara phone har contact name ko ek number se milata hai. Asal mein inmein se har ek pairs ka collection hai, aur har pair mein yeh matter karta hai ki kaun sa item pehle likha gaya. Mathematics mein yahi pattern har jagah hai: '12, 4 ka multiple hai' do numbers ko jodta hai, aur 'triangle $A B C$ , triangle $P Q R$ ke congruent hai' do triangles ko jodta hai.

Yeh chapter is idea ko teen layers mein banata hai. Pehle hum do sets se har possible pair banate hain. Phir hum sirf woh pairs rakhte hain jo kisi rule ko follow karte hain: yeh hai relation. Aakhir mein hum aise relations alag karte hain jinmein har starting element ka theek ek partner hota hai: yeh hain functions, jo baaki mathematics ki working language hain, Chapter 3 ke trigonometric functions se lekar Chapter 12 ke derivatives tak.

2.2 Sets ke Cartesian Products

Ek juice-aur-chai stall teen drinks deta hai, $D = {chai, coffee, lassi}$ , aur har drink do cup sizes mein milti hai, $C = {S, L}$ . Ek customer kitne alag orders de sakta hai? Pehle drink chuno aur phir size. Teen mein se har drink do mein se har size ke saath jaati hai, isliye $3 \times 2 = 6$ orders hain, jinhe Fig. 2.1 ek table mein dikhata hai. Har order ek ordered pair (क्रमित युग्म) hai, jaise $(lassi, S)$ , jiski pehli entry $D$ se ek drink hai aur doosri entry $C$ se ek size.

2/4Har cell ek ordered pair hai: pehle drink, phir size.

Fig. 2.1Har drink har size ke saath pair banati hai: table mein

D \times C

hai.NCERT Fig. 2.1 · p. 24

Ek table jiski teen rows par drinks chai, coffee aur lassi (set D) ke labels hain aur do columns par cup sizes S aur L (set C) ke labels. Pehle step mein chhe cells khaali dashed boxes hain. Doosre step mein har cell apne ordered pair se bhar jaata hai, upar left mein (chai, S) se lekar neeche right mein (lassi, L) tak. Teesre step mein table ke neeche ek line inhe ginti hai: 3 rows guna 2 columns se 6 pairs milte hain. Chauthe step mein cell (lassi, S) highlight hota hai aur table ke neeche ki line kehti hai ki (lassi, S), (S, lassi) ke barabar nahi hai.

Order matter karta hai. $(lassi, S)$ ek small lassi hai, jabki $(S, lassi)$ drink ki jagah size rakh deta hai. Coordinates bhi aise hi kaam karte hain: graph paper par $(2, 5)$ aur $(5, 2)$ alag points hain. Isiliye do ordered pairs tabhi equal maane jaate hain jab unki pehli entries match karein aur doosri entries bhi match karein.

Definition

Cartesian product

Sets $A$ aur $B$ diye hon to unka Cartesian product $A \times B$ (padho ' $A$ cross $B$ ') har woh ordered pair ikattha karta hai jiski pehli entry $A$ se aur doosri entry $B$ se aati hai: $A \times B = {(a, b) : a \in A, b \in B}$ . Agar koi bhi set empty ho, to $A \times B$ bhi empty hota hai.

Example ${1, 2} \times {x} = {(1, x), (2, x)}$ .

Cinema hall apni seats ke liye yahi idea use karta hai. G-14 jaisa ticket pehle row aur phir seat number batata hai. Agar rows A se J tak hain aur har row mein 18 seats hain, to seat labels theek ${A, B, \dots, J} \times {1, 2, \dots, 18}$ ke elements hain, isliye hall mein $10 \times 18 = 180$ seats hain. 14-G jaisa label koi seat hi nahi hai, kyunki pehli entry row honi chahiye.

Cartesian product (कार्तीय गुणन) ki definition se seedhe chaar facts milte hain.

Equality. $(a, b) = (c, d)$ theek tab hota hai jab $a = c$ aur $b = d$ .
Counting. Jab $A$ mein $p$ elements hon aur $B$ mein $q$ elements, to $A \times B$ mein $p q$ elements hote hain, kyunki $p$ pehli entries mein se har ek $q$ doosri entries mein se har ek ke saath pair banati hai. Yahi (2.1) hai.
Infinite sets. Agar $A$ aur $B$ non-empty hain aur unmein se kam se kam ek infinite hai, to $A \times B$ bhi infinite hota hai.
Triples. Teen sets bhi isi tarah judte hain: $A \times B \times C$ mein saare $(a, b, c)$ hote hain jahan $a$ , $A$ se hai, $b$ , $B$ se hai aur $c$ , $C$ se hai. Aise elements ko ordered triplets kehte hain.

product ka size

n (A \times B) = p q

(2.1)

jahan

$p, q$: $n (A)$ aur $n (B)$

Jab dono sets mein real numbers hon, to ordered pair bas coordinate plane ka ek point hota hai. Fig. 2.2 $A = {1, 2, 4}$ aur $B = {1, 3}$ ke liye $A \times B$ plot karta hai, aur phir $B \times A$ . Dono products line $y = x$ mein ek doosre ki mirror image hain aur unmein sirf point $(1, 1)$ common hai. Saare real numbers lene par $𝐑 \times 𝐑$ poora plane hai, aur $𝐑 \times 𝐑 \times 𝐑$ three-dimensional space hai, jo Chapter 11 mein phir aayega.

2/3 $B \times A$ , $y = x$ mein mirror image hai; $(4, 3)$ aur $(3, 4)$ alag points hain.

Fig. 2.2Number sets ka product points ka ek set hota hai.NCERT Fig. 2.3 · p. 25

Dono axes par 0 se 5 tak ka coordinate grid. Pehle step mein chhe gol dots A = {1, 2, 4} aur B = {1, 3} ke liye A × B dikhate hain: points (1, 1), (1, 3), (2, 1), (2, 3), (4, 1) aur (4, 3). Doosre step mein B × A ke liye chhe square markers aate hain, (1, 1), (1, 2), (1, 4), (3, 1), (3, 2) aur (3, 4) par, gol dots fade ho jaate hain, ek dashed mirror line y = x banti hai, aur rings (4, 3) aur (3, 4) ko mark karti hain yeh dikhane ke liye ki entries swap karne se alag point milta hai. Teesre step mein poora grid shaded hota hai aur uska label R × R, poora plane, hota hai.

Example 2.1NCERT Example 1 · p. 26

Barabar pairs

$a$ aur $b$ nikalo agar $(3 a - 2, \frac{b}{2} + 1) = (7, 4)$ .

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.2NCERT Example 2 · p. 26

Factors ko swap karna

Maan lo $A = {p, q}$ aur $B = {1, 2, 3}$ . $A \times B$ aur $B \times A$ ki list banao. Kya dono sets equal hain? Kya dono mein elements ki sankhya same hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.3NCERT Example 3 · p. 26

Intersections aur unions ke saath products

Maan lo $A = {a, b}$ , $B = {1, 2, 3}$ aur $C = {2, 3, 4}$ . (i) $A \times (B \cap C)$ ko $(A \times B) \cap (A \times C)$ se, aur (ii) $A \times (B \cup C)$ ko $(A \times B) \cup (A \times C)$ se compare karo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.4NCERT Example 4 · p. 27

Teen baar coin toss

Ek coin teen baar toss kiya jaata hai aur har result $H$ ya $T$ ke roop mein record hota hai. Saare possible records ke set ko Cartesian product ki tarah likho aur uske elements ki list banao.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.5NCERT Example 5 · p. 27

Number sets ke products

Geometrically describe karo (i) $𝐙 \times 𝐙$ , (ii) $𝐑 \times 𝐑$ aur (iii) $𝐑 \times 𝐑 \times 𝐑$ .

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.6NCERT Example 6 · p. 27

Sets wapas nikalna

Agar $A \times B = {(1, u), (1, v), (1, w), (4, u), (4, v), (4, w)},$ to $A$ aur $B$ nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

4 mein se 1

Agar

n (A) = 5

aur

n (B) = 4

, to

A \times B

mein kitne elements hain?

Answer dekho

Answer 20

(2.1) se,

n (A \times B) = 5 \times 4 = 20

. Sizes ko add karna (9) aam galti hai:

A

ka har element

B

ke har element ke saath pair banata hai.

Practice · 10 questions

Practice 2.1: answers ke saath

Short answerEasyNCERT Exercise 2.1, Q1
$x$ aur $y$ nikalo agar $(\frac{x}{4} + 2, \frac{1}{3} - y) = (\frac{11}{4}, - \frac{2}{3})$ .
Solution dikhao
Answer $x = 3$ , $y = 1$
Pehli entries ko equate karo: $\frac{x}{4} + 2 = \frac{11}{4}$ , isliye $\frac{x}{4} = \frac{3}{4}$ aur $x = 3$ .
Doosri entries ko equate karo: $\frac{1}{3} - y = - \frac{2}{3}$ , isliye $y = \frac{1}{3} + \frac{2}{3} = 1$ .
Quick answerEasyNCERT Exercise 2.1, Q2
Set $A$ mein 4 elements hain aur $B = {a, e, i, o, u}$ . $A \times B$ mein kitne elements hain?
Answer dekho
Answer 20
$n (A \times B) = 4 \times 5 = 20$ .
Short answerEasyNCERT Exercise 2.1, Q3
Maan lo $S = {p, q}$ aur $T = {0, 1, 9}$ . $S \times T$ aur $T \times S$ likho.
Solution dikhao
$S \times T = {(p, 0), (p, 1), (p, 9), (q, 0), (q, 1), (q, 9)} .$
$T \times S = {(0, p), (0, q), (1, p), (1, q), (9, p), (9, q)} .$
Dono mein 6 elements hain, lekin sets alag hain.
4
Short answerEasyNCERT Exercise 2.1, Q4
Batao ki har statement true hai ya false. Har false statement ko sahi karo.
1. Agar $A = {1}$ aur $B = {2, 3}$ , to $A \times B = B \times A$ .
  SahiGalat
  Solution dikhao
  False. $A \times B = {(1, 2), (1, 3)}$ lekin $B \times A = {(2, 1), (3, 1)}$ . Sahi version: $A \times B$ aur $B \times A$ mein elements ki sankhya same hai, lekin yeh alag sets hain.
2. Finite sets $A$ aur $B$ ke liye, $n (A \times B) = n (B \times A)$ .
  SahiGalat
  Solution dikhao
  True: dono $n (A) \cdot n (B)$ ke barabar hain.
3. Agar $A = {4, 5}$ , to $A \times \emptyset$ mein do elements hain.
  SahiGalat
  Solution dikhao
  False. Pair banane ke liye koi doosri entry hai hi nahi, isliye $A \times \emptyset = \emptyset$ , jismein koi element nahi hai.
Short answerEasyNCERT Exercise 2.1, Q5
Agar $A = {0, 1}$ , to $A \times A \times A$ ki list banao. Iske kitne triplets ki entries ka sum 2 hai?
Solution dikhao
Answer8 triplets; unmein se 3 ka sum 2 hai.
$A \times A \times A$ mein $2^{3} = 8$ triplets hain: $(0, 0, 0)$ , $(0, 0, 1)$ , $(0, 1, 0)$ , $(0, 1, 1)$ , $(1, 0, 0)$ , $(1, 0, 1)$ , $(1, 1, 0)$ , $(1, 1, 1)$ .
Entries ka sum 2 tab hota hai jab theek do entries 1 hon: $(0, 1, 1)$ , $(1, 0, 1)$ aur $(1, 1, 0)$ , yaani 3 triplets.
Short answerEasyNCERT Exercise 2.1, Q6
Agar $A \times B = {(2, m), (2, n), (5, m), (5, n), (7, m), (7, n)},$ to $A$ aur $B$ nikalo.
Solution dikhao
Answer $A = {2, 5, 7}$ , $B = {m, n}$
Pehli entries: $A = {2, 5, 7}$ . Doosri entries: $B = {m, n}$ . Check: $3 \times 2 = 6$ pairs.
7
Short answerMediumNCERT Exercise 2.1, Q7
Maan lo $A = {p, q}$ , $B = {1, 3, 5}$ aur $C = {3, 5, 7}$ .
1. Verify karo ki $A \times (B \cap C) = (A \times B) \cap (A \times C)$ .
  Solution dikhao
  $B \cap C = {3, 5}$ , isliye left side yeh hai: ${(p, 3), (p, 5), (q, 3), (q, 5)} .$
  Koi pair $A \times B$ aur $A \times C$ dono mein theek tab hota hai jab uski doosri entry $B$ mein bhi ho aur $C$ mein bhi, yaani 3 ya 5 ho. Isse wahi chaar pairs milte hain, isliye dono sides equal hain.
2. $E = {p, q, r}$ aur $D = {1, 3, 5, 7, 9}$ ke saath dikhao ki $A \times C \subset E \times D$ .
  Solution dikhao
  Koi bhi $(x, y) \in A \times C$ lo. Tab $x \in A \subset E$ aur $y \in C \subset D$ , isliye $(x, y) \in E \times D$ . Isliye $A \times C \subset E \times D$ ; jaise $(r, 1) \in E \times D$ , $A \times C$ mein nahi hai, isliye inclusion strict hai.
Short answerEasyNCERT Exercise 2.1, Q8
$A = {a}$ aur $B = {1, 2, 3}$ ke liye $A \times B$ ke elements ki list banao, batao ki iske kitne subsets hain, aur sabko likho.
Solution dikhao
$A \times B = {(a, 1), (a, 2), (a, 3)}$ , 3 elements wala set hai, isliye iske $2^{3} = 8$ subsets hain:
$\emptyset$ , ${(a, 1)}$ , ${(a, 2)}$ , ${(a, 3)}$ , ${(a, 1), (a, 2)}$ , ${(a, 1), (a, 3)}$ , ${(a, 2), (a, 3)}$ aur khud $A \times B$ .
Short answerMediumNCERT Exercise 2.1, Q9
Sets $A$ aur $B$ mein $n (A) = 2$ aur $n (B) = 3$ . Pairs $(p, u)$ , $(q, v)$ aur $(p, w)$ , $A \times B$ mein hain, jahan $p \neq q$ aur $u$ , $v$ , $w$ alag-alag hain. $A$ aur $B$ nikalo.
Solution dikhao
Answer $A = {p, q}$ , $B = {u, v, w}$
Har pehli entry $A$ mein hai, isliye $p, q \in A$ . Yeh alag hain aur $n (A) = 2$ , isliye $A = {p, q}$ .
Har doosri entry $B$ mein hai, isliye $u, v, w \in B$ . Yeh alag hain aur $n (B) = 3$ , isliye $B = {u, v, w}$ .
Short answerMediumNCERT Exercise 2.1, Q10
$A \times A$ ke nau elements mein se do $(2, - 1)$ aur $(5, 2)$ hain. $A$ nikalo, phir $A \times A$ ke baaki elements ki list banao.
Solution dikhao
$n (A)^{2} = 9$ , isliye $n (A) = 3$ . Entries $2$ , $- 1$ aur $5$ sab $A$ mein honi chahiye, aur yeh pehle se teen elements hain, isliye $A = {- 1, 2, 5}$ .
Baaki saat elements $(- 1, - 1)$ , $(- 1, 2)$ , $(- 1, 5)$ , $(2, 2)$ , $(2, 5)$ , $(5, - 1)$ aur $(5, 5)$ hain.

2.3 Relations

Chaar dost, Asha, Ravi, Meena aur Kabir, aur chaar games, cricket, kabaddi, chess aur hockey, $4 \times 4 = 16$ possible (student, game) pairs dete hain. Aam taur par inmein se sirf kuch pairs mein hi humari dilchaspi hoti hai, jaise woh pairs jinmein student sach mein woh game khelta hai. Aise kisi rule se pairs chunne par ek relation (संबंध) banta hai. Fig. 2.3 ise ek arrow diagram ki tarah banata hai, jismein har student se har us game tak arrow hai jo woh student khelta hai.

3/3Range un games ka set hai jin tak arrow pahunchta hai: hockey codomain mein hai lekin range mein nahi.

Fig. 2.3Ek relation, usse padhe jaane wale teen sets.NCERT Fig. 2.4 · p. 28

Do ovals: left wale, set A, mein students Asha, Ravi, Meena aur Kabir hain; right wale, set B, mein games cricket, kabaddi, chess aur hockey hain. Chaar arrows hain: Asha se cricket, Asha se chess, Ravi se kabaddi aur Kabir se cricket. Doosre step mein Asha, Ravi aur Kabir domain ke roop mein box hote hain, jabki Meena is note ke saath fade hoti hai ki uska koi arrow nahi hai. Teesre step mein cricket, kabaddi aur chess range ke roop mein box hote hain, jabki hockey is note ke saath fade hota hai ki us tak koi arrow nahi pahunchta, halaanki woh ab bhi codomain B mein hai.

Definition

Relation

Non-empty sets $A$ aur $B$ diye hon to $A \times B$ ke kisi bhi subset $R$ ko $A$ se $B$ tak ka relation kehte hain. Aam taur par dono entries ko jodne wala ek rule decide karta hai ki kaun se pairs $R$ mein hain. Jab $(a, b) \in R$ , to hum $b$ ko $a$ ki image kehte hain.

Example ${1, 2, 3}$ se ${1, 4, 9}$ tak, rule 'doosri entry pehli ka square hai' se $R = {(1, 1), (2, 4), (3, 9)}$ milta hai.

Definition

Domain

$A$ se $B$ tak ke relation $R$ ka domain, $R$ ke pairs ki saari pehli entries ka set hai. Yeh $A$ ka subset hai.

Definition

Codomain

$A$ se $B$ tak ke relation ka codomain poora set $B$ hai, chahe $B$ ka har element use ho ya nahi.

Definition

Range

$R$ ki range uske pairs ki doosri entries (images) ikatthi karti hai. Yeh hamesha codomain ka subset hoti hai.

Inhe Fig. 2.3 se padho. Relation yeh hai: ${(Asha, cricket), (Asha, chess), (Ravi, kabaddi), (Kabir, cricket)} .$ Iska domain (प्रांत) ${Asha, Ravi, Kabir}$ hai: Meena inmein se koi game nahi khelti, isliye woh kabhi pehli entry nahi banti. Iska codomain (सहप्रांत) chaaron games ka set hai, lekin iski range (परिसर) sirf ${cricket, kabaddi, chess}$ hai, kyunki hockey koi nahi khelta. Asha ki do images (प्रतिबिंब) hain, cricket aur chess.

Relation ko dikhane ke teen standard tareeke hain. Roster form pairs ki list deta hai. Set-builder form rule batata hai, jaise $R = {(x, y) : y = x^{2}, x \in A, y \in B}$ . Arrow diagram pairs ko draw karta hai. Kisi set $A$ se usi set $A$ tak ke relation ko $A$ par relation kehte hain.

Example 2.7NCERT Example 7 · p. 28

Ek hi set ke andar doubling

Maan lo $A = {1, 2, 3, \dots, 9}$ aur $R = {(x, y) : y = 2 x, x, y \in A}$ , $A$ par ek relation hai. (i) $R$ ko roster form mein likho aur uske arrow diagram ko describe karo. (ii) $R$ ka domain, codomain aur range nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.8NCERT Example 8 · p. 29

Arrows se rule padhna

$P = {6, 8, 9}$ se $Q = {2, 3, 4, 5}$ tak ke ek arrow diagram mein theek yeh arrows hain: $6 \to 2$ , $6 \to 3$ , $8 \to 2$ , $8 \to 4$ aur $9 \to 3$ . Relation ko set-builder form aur roster form mein likho, aur uska domain aur range nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

$A$ se $B$ tak kitne relations define kiye ja sakte hain? Relation bas $A \times B$ ka ek subset hai, isliye hum subsets gin rahe hain. $A$ mein $p$ aur $B$ mein $q$ elements hon to product $A \times B$ mein $p q$ pairs hain, aur har pair ya to relation mein hai ya bahar: har pair ke liye do choices, jo ek doosre se independent hain.

relations ki sankhya

N = 2^{p q}

(2.2)

jahan

$N$: $A \times B$ ke saare subsets ki ginti
$p, q$: $n (A)$ aur $n (B)$

Is ginti mein do extreme cases bhi shaamil hain: empty relation $\emptyset$ , jismein kuch bhi related nahi hai, aur poora $A \times B$ , jismein sab kuch related hai.

Example 2.9NCERT Example 9 · p. 29

Relations ginna

Maan lo $A = {x, y}$ aur $B = {1, 2, 3}$ . (i) $A$ se $B$ tak kitne relations hain? (ii) Unmein se kitne pair $(x, 1)$ rakhte hain?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

4 mein se 1

A = {1, 2, \dots, 8}

par maan lo

R = {(x, y) : x + y = 10}

R

ki range kya hai?

{1, 2, \dots, 8}

{5}

{2, 3, \dots, 8}

{10}

Answer dekho

Answer (C) ${2, 3, \dots, 8}$

Pairs

(2, 8), (3, 7), \dots, (8, 2)

hain, isliye doosri entries 2 se 8 tak jaati hain. Element 1 range mein nahi hai: uska partner 9 hota, jo

A

mein nahi hai. Codomain poora

A

hai; range usse chhoti hai.

Practice · 9 questions

Practice 2.2: answers ke saath

Short answerEasyNCERT Exercise 2.2, Q1
Maan lo $A = {1, 2, 3, \dots, 20}$ aur $R = {(x, y) : y = 4 x, x, y \in A}$ . $R$ ka domain, codomain aur range nikalo.
Solution dikhao
$4 x \leq 20$ sirf $x \leq 5$ ke liye, isliye $R = {(1, 4), (2, 8), (3, 12), (4, 16), (5, 20)} .$
Domain $= {1, 2, 3, 4, 5}$ , codomain $= A$ aur range $= {4, 8, 12, 16, 20}$ .
Short answerEasyNCERT Exercise 2.2, Q2
$𝐍$ par $R = {(x, 3 x - 1) : x \in 𝐍, x \leq 4}$ lo. $R$ ki list banao aur uska domain aur range batao.
Solution dikhao
$x = 1, 2, 3, 4$ se $y = 2, 5, 8, 11$ milte hain. Isliye $R = {(1, 2), (2, 5), (3, 8), (4, 11)}$ , jiska domain ${1, 2, 3, 4}$ aur range ${2, 5, 8, 11}$ hai.
Short answerMediumNCERT Exercise 2.2, Q3
Maan lo $A = {2, 3, 4, 7}$ aur $B = {5, 6, 10}$ . ' $x + y$ , 3 ka multiple hai' se define kiye gaye $A$ se $B$ tak ke relation $R$ ko roster form mein likho.
Solution dikhao
Saare 12 sums check karo. 3 ke multiples $2 + 10 = 12$ , $3 + 6 = 9$ , $4 + 5 = 9$ aur $7 + 5 = 12$ hain; baaki aath sums (7, 8, 8, 13, 10, 14, 13, 17) nahi hain.
Isliye $R = {(2, 10), (3, 6), (4, 5), (7, 5)}$ .
Short answerEasyNCERT Exercise 2.2, Q4
$P = {10, 12, 14}$ se $Q = {5, 6, 7, 8}$ tak ke ek arrow diagram mein arrows $10 \to 5$ , $12 \to 6$ aur $14 \to 7$ hain. Relation ko set-builder form aur roster form mein likho, aur uska domain aur range nikalo.
Solution dikhao
Har arrow apne starting number ko aadha karta hai, isliye rule $y = \frac{x}{2}$ hai.
Set-builder form mein: $R = {(x, y) : y = \frac{x}{2}, x \in P, y \in Q}$ . Roster form mein: $R = {(10, 5), (12, 6), (14, 7)}$ .
Domain $= {10, 12, 14}$ ; range $= {5, 6, 7}$ (8, $Q$ mein hai lekin us tak koi arrow nahi pahunchta).
5
Short answerMediumNCERT Exercise 2.2, Q5
$A = {2, 3, 4, 6, 9}$ par maan lo $R = {(a, b) : a \neq b and b is a multiple of a}$ .
1. $R$ ko roster form mein likho.
  Solution dikhao
  $A$ ke andar multiples: 2 ke 4 aur 6; 3 ke 6 aur 9; 4, 6 aur 9 ka khud ke alawa koi nahi. Isliye $R = {(2, 4), (2, 6), (3, 6), (3, 9)}$ .
2. $R$ ka domain nikalo.
  Solution dikhao
  Domain $= {2, 3}$ .
3. $R$ ki range nikalo.
  Solution dikhao
  Range $= {4, 6, 9}$ .
Short answerEasyNCERT Exercise 2.2, Q6
$R = {(x, x^{2} - 1) : x \in {- 2, - 1, 0, 1, 2}}$ ka domain aur range nikalo.
Solution dikhao
Pairs $(- 2, 3)$ , $(- 1, 0)$ , $(0, - 1)$ , $(1, 0)$ aur $(2, 3)$ hain.
Domain $= {- 2, - 1, 0, 1, 2}$ ; range $= {- 1, 0, 3}$ (repeat hone wali values ek hi baar likhi jaati hain).
Short answerEasyNCERT Exercise 2.2, Q7
Relation $R = {(p, 2^{p})}$ , 8 se chhote har prime $p$ ko 2 ki ek power se jodta hai. $R$ ko roster form mein likho.
Solution dikhao
8 se chhote primes 2, 3, 5 aur 7 hain, isliye $R = {(2, 4), (3, 8), (5, 32), (7, 128)}$ .
Quick answerEasyNCERT Exercise 2.2, Q8
${p, q, r}$ se ${1, 2, 3}$ tak kitne relations define kiye ja sakte hain?
Answer dekho
Answer 512
$A \times B$ mein 9 pairs hain, isliye $2^{9} = 512$ relations hain.
Short answerMediumNCERT Exercise 2.2, Q9
Maan lo $R = {(a, b) : a, b \in 𝐍 and b = a + 2}$ . $R$ ka domain aur range nikalo.
Solution dikhao
Har natural number $a$ ka partner $a + 2 \in 𝐍$ hai, isliye domain $𝐍$ hai.
Doosri entries $3, 4, 5, \dots$ hain. Numbers 1 aur 2 kabhi nahi aate, kyunki unke liye $a = - 1$ ya $a = 0$ chahiye hoga. Isliye range ${3, 4, 5, \dots} = 𝐍 - {1, 2}$ hai.

2.4 Functions

Kuch relations ki ek khaas property hoti hai. India mein har SIM card theek ek person ke naam par registered hota hai, halaanki ek person ke paas do ya teen SIMs ho sakte hain. Isliye relation 'SIM → registered owner' har SIM ko theek ek person tak bhejta hai, jabki 'person → SIM' ek person ko kai numbers tak bhej sakta hai. Pehli tarah ke relations ek machine ki tarah kaam karte hain jo har input ko ek pakka output banati hai. Inhe functions (फलन) kehte hain; map aur mapping shabdon ka bhi yahi matlab hai.

Definition

Function

Maan lo $A$ aur $B$ non-empty sets hain. Relation $f \subseteq A \times B$ , $A$ se $B$ tak ka function tab hai jab $A$ ka har element $f$ ke theek ek pair ki pehli entry ho. Tab hum $f : A \to B$ likhte hain, aur pair $(a, b)$ ke liye $f (a) = b$ .

Example $f : {1, 2, 3} \to {0, 1}$ jismein $f (1) = 1$ , $f (2) = 0$ , $f (3) = 1$ .

$f (a) = b$ ko hum ' $f$ of $a$ equals $b$ ' padhte hain: output $b$ , $a$ ki image hai, jabki $a$ , $b$ ki ek preimage (पूर्व प्रतिबिंब) hai. Definition mein do alag conditions hain, aur Fig. 2.4 har ek ko test karta hai. Pehli, $A$ ka koi element chhoota nahi ja sakta, isliye $f$ ka domain poora $A$ hai. Doosri, $A$ ke kisi element ki do images nahi ho sakti, isliye $f$ ke kisi do pairs ki pehli entry same nahi hoti. Do alag inputs ka output same ho sakta hai, aur $B$ ke kuch elements tak koi arrow na pahunche, yeh bhi chalta hai: dono se rule nahi tootta.

2/3Ab 1 ki do images hain, $a$ aur $b$ : function nahi hai.

Fig. 2.4

A

ke har element se ek arrow, na zyada na kam.

Do ovals, A mein 1, 2 aur 3 aur B mein a, b, c aur d. Pehle step mein arrows 1 se a, 2 se c aur 3 se c hain, aur tick wala label kehta hai ki yeh function hai: 2 aur 3 image c share karte hain aur d tak kuch nahi pahunchta, jo allowed hai. Doosre step mein 1 se b tak ek extra dashed arrow aata hai, 1 par ring lagti hai, aur label kehta hai ki 1 ki do images hain, isliye yeh function nahi hai. Teesre step mein extra arrow hat jaata hai lekin 3 se aane wala arrow bhi hata diya jaata hai, 3 par ring lagti hai, aur label kehta hai ki 3 ki koi image nahi hai, isliye yeh bhi function nahi hai.

Section 2.3 ke relations ko is par test karo. Example 2.7 mein numbers 5 se 9 ki koi image nahi hai, isliye woh relation $A$ par function nahi hai. Example 2.8 mein number 6 ki do images hain, 2 aur 3. Example 2.9 ke 64 relations mein se sirf woh functions hain jinmein $x$ se shuru hone wala theek ek pair aur $y$ se shuru hone wala theek ek pair ho, aur aise $3 \times 3 = 9$ hain.

Jab $A$ aur $B$ real numbers ke sets hon, to relation plane mein points ka ek set hota hai aur test visual ban jaata hai. Function kabhi ek hi vertical line par do points nahi rakhta, aur uske domain ke har $x$ ke upar (ya neeche) ek point hota hai. Simulation mein try karo.

SimulationKya yeh ab bhi function hai?

Dono axes par 0 se 8 tak ka coordinate grid, jismein chaar draggable points A (1, 3), B (2, 5), C (4, 5) aur D (6, 2) hain, har ek apne coordinates dikhata hai aur whole numbers par snap hota hai. Saath mein yeh relation {(1, 3), (2, 5), (4, 5), (6, 2)} banate hain. Ek point drag karne se ek pair badalta hai, isliye tum same pehli entry wale do pairs bana sakte ho aur dekh sakte ho ki relation kab function hona band karta hai.

Fig. 2.5Har point relation ka ek pair

(x, y)

hai. Points ko drag karo aur dekho kab do points ek vertical line par aate hain.

Try karo

$D$ ko $(4, 1)$ par drag karo, $C$ ke theek neeche. Ab kis input ki do images hain?
$B$ aur $C$ ka $y$ -coordinate 5 same hai. Kya isse relation function hona band ho jaata hai?
Points ko aise lagao ki domain ${1, 2, 3, 4}$ ho aur relation ab bhi function rahe.

Example 2.10NCERT Example 10 · p. 31

Natural numbers ke squares

Maan lo $R = {(x, y) : y = x^{2}, x, y \in 𝐍}$ . $R$ ka domain, codomain aur range nikalo. Kya $R$ , $𝐍$ se $𝐍$ tak function hai?

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.11NCERT Example 11 · p. 31

Kaun si lists functions hain?

Decide karo ki har relation ${1, 2, 3, 4}$ se $𝐍$ tak function hai ya nahi, aur reason do. (i) ${(1, 5), (2, 5), (3, 5), (4, 5)}$ (ii) ${(1, 2), (2, 3), (3, 4)}$ (iii) ${(1, 1), (2, 4), (3, 9), (2, 6), (4, 16)}$

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Definition

Real function

Real function woh hai jiska domain aur range dono $𝐑$ ke andar hon. Agar sirf range ka $𝐑$ mein hona zaroori ho, to function ko real-valued kehte hain.

Example $f (x) = \sqrt{x}$ , domain $[0, \infty)$ ke saath.

Is book ke zyadatar functions formula se diye gaye real functions (वास्तविक फलन) hain. Jab sirf formula diya ho, to hum uska domain real numbers ka woh sabse bada set lete hain jiske liye formula ka matlab banta hai. Jaise, $\frac{1}{x - 2}$ ka domain $𝐑 - {2}$ hai aur $\sqrt{x + 1}$ ka domain $[- 1, \infty)$ hai.

Example 2.12NCERT Example 12 · p. 31

Auto fares ki ek table

Ek town mein auto-rickshaw fixed ₹25 aur har kilometre ke ₹15 leta hai, isliye $x$ km ka fare $f (x) = 15 x + 25$ rupees hai, jahan $x \in 𝐍$ . $x = 1$ se $6$ tak $f (x)$ ki table banao.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2.4.1 Kuch functions aur unke graphs

Real function $f$ ka graph points $(x, f (x))$ ka set hai, jo $𝐑 \times 𝐑$ ka subset hai. Function ka domain, range aur uska behaviour dekhne ka sabse tez tareeka aksar ek sketch hota hai. Yeh rahi woh saat families jo tum school mathematics mein aage hamesha use karoge.

(i) Identity function. Function $f : 𝐑 \to 𝐑$ , $f (x) = x$ , har input ko bina badle wapas deta hai (तत्समक फलन). Iska graph line $y = x$ hai, jo origin se guzarti hai aur har axis ke saath $45^{\circ}$ ka angle banati hai. Iska domain aur range dono $𝐑$ hain.

(ii) Constant functions. Ek fixed number $c$ ke liye function $f : 𝐑 \to 𝐑$ , $f (x) = c$ , har input ko same output deta hai (अचर फलन). Iska graph height $c$ par ek horizontal line hai. Domain $𝐑$ hai, lekin range single-element set ${c}$ hai. Fig. 2.6 dono graphs dikhata hai.

2/2 $y = 2$ : har input se 2 milta hai, isliye range ${2}$ hai.

Fig. 2.6Identity aur constant functions dono straight lines hain.NCERT Fig. 2.8 · p. 32

Minus 4 se 4 tak ke axes. Pehle step mein line y = x origin se 45 degrees par upar uthti hai, marked points (−3, −3) aur (2, 2) ke saath; labels iska domain aur range saare real numbers batate hain. Doosre step mein identity line fade hoti hai aur horizontal line y = 2 aati hai, x = −3 aur x = 1 par marked points ke saath, dono height 2 par; ab labels domain saare real numbers aur range sirf ek value 2 batate hain.

(iii) Polynomial functions. Yeh $x$ ki whole-number powers se bante hain (बहुपद फलन).

Definition

Polynomial function

Function $f : 𝐑 \to 𝐑$ polynomial function hai agar kisi non-negative integer $n$ aur real numbers $a_{0}, a_{1}, \dots, a_{n}$ ke liye $f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}$ ho.

Example $x^{5} - 3 x + 7$ aur $\frac{2}{3} x^{2}$ .

polynomial function

\begin{aligned} f (x) & = a_{0} + a_{1} x \\ + a_{2} x^{2} + \dots \\ + a_{n} x^{n} \end{aligned}

(2.3)

Coefficients koi bhi real numbers ho sakte hain, lekin powers $0, 1, 2, \dots$ hi honi chahiye. Isliye $g (x) = π x^{3} - 5 x + \frac{1}{2}$ ek polynomial function hai, jabki $h (x) = x^{1 / 2} + 1$ aur $k (x) = x + \frac{1}{x}$ nahi hain, kyunki inmein powers $\frac{1}{2}$ aur $- 1$ hain.

Example 2.13NCERT Example 13 · p. 33

Squaring function

$f : 𝐑 \to 𝐑$ , $f (x) = x^{2}$ ke liye $x = - 2$ , $- 1.5$ , $- 1$ , $- 0.5$ , $0$ , $0.5$ , $1$ , $1.5$ , $2$ par $f (x)$ nikalo. Graph sketch karo aur domain aur range batao.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2/3Inse guzarta smooth curve parabola $y = x^{2}$ hai.

Fig. 2.7Alag shapes wale do power functions.NCERT Fig. 2.10 · p. 33

Axes jinmein x minus 3 se 3 tak aur y minus 8 se 8 tak hai (vertical scale compressed hai). Pehle step mein nau dots x = −2 se 2 tak x squared ki table values mark karte hain: 4, 2.25, 1, 0.25, 0, 0.25, 1, 2.25, 4. Doosre step mein ek smooth U-shaped parabola dots se guzarta hai; labels batate hain ki uski range 0 se upar hai aur woh y-axis mein khud ki mirror image hai. Teesre step mein parabola fade hota hai aur cubic curve y = x cubed aata hai, jo x = −2 par −8 tak girta hai, origin par flat hota hua guzarta hai aur x = 2 par 8 tak uthta hai; labels batate hain ki uski range saare real numbers hai aur woh har jagah upar uthta hai.

Example 2.14NCERT Example 14 · p. 34

Cubing function

Example 2.13 wale same inputs use karke $f : 𝐑 \to 𝐑$ , $f (x) = x^{3}$ ka graph sketch karo, aur uski range nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

(iv) Rational functions. Ek polynomial function ko doosre se divide karne par rational function (परिमेय फलन) milta hai.

Definition

Rational function

$\frac{p (x)}{q (x)}$ form ka function, jahan $p$ aur $q$ polynomial functions hain. Yeh har jagah defined hai jahan $q (x) \neq 0$ .

Example $\frac{x + 1}{x - 2}$ , har $x \neq 2$ ke liye defined.

Example 2.15NCERT Example 15 · p. 34

Reciprocal function

Maan lo $x \in 𝐑 - {0}$ ke liye $f (x) = \frac{1}{x}$ . $x = - 4$ , $- 2$ , $- 1$ , $- 0.5$ , $- 0.25$ , $0.25$ , $0.5$ , $1$ , $2$ , $4$ par $f (x)$ nikalo, graph sketch karo aur domain aur range nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2/3Inhe jodo: ek branch $x > 0$ ke liye, ek $x < 0$ ke liye.

Fig. 2.8Reciprocal function ki do branches hain.NCERT Fig. 2.12 · p. 35

Minus 4 se 4 tak ke axes. Pehle step mein das dots 1/x ki table values mark karte hain, (−4, −0.25) se (4, 0.25) tak, aur axes se door wale points (0.25, 4) aur (−0.25, −4) par hain. Doosre step mein do smooth branches dots ko jodti hain: ek first quadrant mein aur ek third mein, dono x ke right jaane par neeche girti hain. Teesre step mein dono axes dashed bante hain aur unpar label hai: x kabhi 0 nahi aur y kabhi 0 nahi, isliye graph kisi bhi axis se nahi milta.

(v) Modulus function. Function $f (x) = ∣ x ∣$ number line par 0 se $x$ ki distance deta hai, isliye yeh kabhi negative nahi hota (मापांक फलन). Do pieces mein:

modulus function

∣ x ∣ = {\begin{cases} x, & x \geq 0 \\ - x, & x < 0 \end{cases}

(2.4)

Modulus function banane ke liye line $y = x$ se shuru karo aur $x$ -axis ke neeche wale part ko upar flip kar do (Fig. 2.9). Result ek V hai jiska corner origin par hai. Domain $𝐑$ hai aur range $[0, \infty)$ hai.

3/3Opposite inputs ke outputs equal hain: $∣ - 3 ∣ = ∣ 3 ∣ = 3$ .

Fig. 2.9

y = x

ko fold karne se

y = ∣ x ∣

milta hai.NCERT Fig. 2.13 · p. 35

Minus 4 se 4 tak ke axes. Pehle step mein poori line y = x banti hai. Doosre step mein line ka y-axis ke left wala half x-axis ke about upar flip hota hai, aur purani jagah ek halki dashed copy chhod deta hai; ab solid graph ek V hai jiska corner origin par hai, label y = |x|. Teesre step mein (−3, 3) aur (3, 3) par dots ek dashed level line se jude hain, aur labels batate hain ki |−3| = |3| = 3 aur range 0 se upar hai.

(vi) Signum function. Signum Latin mein 'sign' ko kehte hain, aur signum function apne input ka sirf sign batata hai (चिह्न फलन):

signum function

sgn x = {\begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ - 1, & x < 0 \end{cases}

(2.5)

Iska graph (Fig. 2.10) heights 1 aur $- 1$ par do horizontal rays se banta hai, jinka $y$ -axis par wala end hollow hai, aur ek single point $(0, 0)$ se. Domain $𝐑$ hai aur range ${- 1, 0, 1}$ hai. $x \neq 0$ ke liye tum $sgn x = \frac{∣ x ∣}{x}$ bhi likh sakte ho.

3/3 $sgn 0 = 0$ : origin par bhara hua dot graph poora karta hai.

Fig. 2.10Signum function sirf sign rakhta hai.NCERT Fig. 2.14 · p. 36

Axes jinmein x minus 4 se 4 tak aur y minus 3 se 3 tak hai. Pehle step mein height 1 par ek horizontal ray (0, 1) ke hollow circle se right ki taraf jaati hai, label: x positive se 1 milta hai. Doosre step mein height −1 par ek ray (0, −1) ke hollow circle se left ki taraf jaati hai, label: x negative se −1 milta hai. Teesre step mein origin par ek bhara hua dot lagta hai, label: x = 0 se 0 milta hai, aur range −1, 0 aur 1 likhi jaati hai.

(vii) Greatest integer function. Greatest integer function $f (x) = [x]$ woh sabse bada integer deta hai jo $x$ se kam ya uske barabar ho (महत्तम पूर्णांक फलन). Isliye $[3.7] = 3$ , $[5] = 5$ aur $[- 0.4] = - 1$ . Negative inputs ke liye neeche round karna zero se door le jaata hai.

greatest integer function

\begin{array}{c} [x] = n \\ jab n \leq x < n + 1 \end{array}

(2.6)

jahan

$n$: ek integer

Har interval $[n, n + 1)$ par value $n$ par rehti hai, aur agle integer par 1 se upar jump karti hai. Graph ek staircase hai (Fig. 2.11): har step apna left end (bhara dot) shaamil karta hai lekin right end (hollow dot) nahi. Domain $𝐑$ hai aur range integers ka set $𝐙$ hai.

2/2Har step $[n, n + 1)$ cover karta hai: left end par bhara, right end par hollow.

Fig. 2.11Greatest integer function ek staircase hai.NCERT Fig. 2.15 · p. 36

Minus 3 se 4 tak ke axes. Pehle step mein x-axis par do inputs mark hote hain, 2.6 aur −1.3, aur har ek se uske output level tak dashed arrows jaate hain: 2.6, 2 par jaata hai aur −1.3, −2 par; labels zor dete hain ki function hamesha neeche round karta hai. Doosre step mein poori staircase aati hai: heights −3 se 3 tak horizontal steps, har ek left end par bhare dot se shuru aur right end par hollow circle par khatam; labels range integers batate hain aur yeh bhi ki bhara end step mein shaamil hai.

InteractiveFunction lab

Do modes. Graphs mode mein buttons identity, constant, square, cube, reciprocal, modulus, signum ya greatest integer function chunte hain; graph −5 se 5 tak ke grid par banta hai, ek slider input x ko chalata hai aur point (x, f(x)) mark karta hai, aur ek panel domain aur range batata hai (constant c ke liye ek slider ke saath). Algebra mode mein do menus x, x², x + 1, 2x − 3, |x|, √x aur 1/x mein se f aur g chunte hain, aur buttons f + g, f − g, f · g, f ÷ g ya k · f chunte hain (k ke liye ek slider ke saath). f aur g ke graphs halke aur result gaadha banta hai, hataaye gaye points par hollow circles ke saath, aur panel f(x), g(x), result aur result ka domain deta hai, jismein g ke woh zeros bhi shaamil hain jinhe quotient ko chhodna padta hai.

Fig. 2.12Saat families ka graph banao, ek input ko

x

-axis par chalao, aur do functions ko add, subtract, multiply ya divide karke jodo.

Quick check

4 mein se 1

Har function ko uski range se match karo.

Left mein koi item tap karo, phir right mein uska match.

Answer dekho

Answer

$x^{2}$ : $[0, \infty)$
$x^{3}$ : $𝐑$
$sgn x$ : ${- 1, 0, 1}$
$\frac{1}{x}$ : $𝐑 - {0}$

Squares kabhi negative nahi hote; cubes har real value lete hain; signum sirf teen values leta hai;

\frac{1}{x}

, 0 ke alawa har value leta hai.

2.4.2 Real functions ka algebra

Real functions ko lagbhag numbers ki tarah hi joda ja sakta hai, ek baar mein ek input par. Same domain $X \subseteq 𝐑$ wale do real functions $f$ aur $g$ lo. $(f + g) (x)$ nikalne ke liye $f (x)$ aur $g (x)$ nikalo aur unhe add karo. Yahi pointwise idea difference, product, quotient aur scalar multiple $α f$ ko define karta hai, jahan scalar $α$ ek real number hai.

Table 2.1Real functions

f, g : X \to 𝐑

ka algebra, har

x \in X

ke liye (

α

ek real number hai).

Combination	$x$ par value
sum	$(f + g) (x) = f (x) + g (x)$
difference	$(f - g) (x) = f (x) - g (x)$
scalar multiple	$(α f) (x) = α f (x)$
product	$(f g) (x) = f (x) g (x)$
quotient	$(\frac{f}{g}) (x) = \frac{f (x)}{g (x)}$ , jahan $g (x) \neq 0$

Quotient ke liye ek extra condition chahiye: $(\frac{f}{g}) (x)$ sirf wahan exist karta hai jahan $g (x) \neq 0$ , isliye iska domain $X$ hai jismein se $g$ ke zeros hata diye gaye hon. Agar $f$ aur $g$ ke domains shuru se alag hon, to combinations us part par defined hote hain jo dono mein common ho. Product $f g$ ko pointwise multiplication kehte hain, taaki ise composition se alag rakha ja sake, yaani ek function ke baad doosra function lagana, jo tum Class 12 mein padhoge.

Graph par do functions ko add karne ka matlab hai heights ko ek ke upar ek rakhna (Fig. 2.13). Har $x$ par $f + g$ ka point, $f$ ke point se $g (x)$ distance upar hota hai, ya neeche jab $g (x)$ negative ho.

3/3Har $x$ par aisa karne se $f + g$ ka graph banta hai.

Fig. 2.13Functions add karne ke liye har

x

par unki heights add karo.

Axes jinmein x minus 4 se 4 tak aur y minus 2 se 7 tak hai. Pehle step mein straight line f(x) = x/2 + 1 aur dashed parabola g(x) = x²/4 bante hain. Doosre step mein do stacked bars aate hain: x = 2 par height f(2) = 2 ka bar aur uske upar height g(2) = 1 ka bar, jo 3 tak pahunchta hai; x = −3 par f(−3) = −0.5 tak neeche ek chhota bar aur wahan se shuru hone wala height g(−3) = 2.25 ka bar, jo 1.75 tak pahunchta hai. Dots tops ko mark karte hain. Teesre step mein f + g ka curve, jo x²/4 + x/2 + 1 ke barabar hai, dono dots se guzarta hai jabki f aur g fade ho jaate hain.

Example 2.16NCERT Example 16 · p. 37

Chaar combinations

Maan lo $f (x) = x^{2} - 1$ aur $g (x) = x + 3$ , $𝐑$ par real functions hain. $f + g$ , $f - g$ , $f g$ aur $\frac{f}{g}$ ke formulas likho.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.17NCERT Example 17 · p. 37

Mix mein ek square root

Maan lo $X = [0, \infty)$ par $f (x) = \sqrt{x}$ aur $g (x) = 3 - x$ . $f + g$ , $f - g$ , $f g$ aur $\frac{f}{g}$ ko formulas ki tarah likho, aur har ek ka domain do.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Quick check

3 mein se 1

Agar

f (x) = 2 x + 1

aur

g (x) = x^{2}

, to

(f - g) (3)

nikalo.

Answer dekho

Answer -2

f (3) = 7

aur

g (3) = 9

, isliye

(f - g) (3) = 7 - 9 = - 2

. Order par dhyaan do:

g - f

+ 2

milega.

Practice · 5 questions

Practice 2.3: answers ke saath

1
Short answerEasyNCERT Exercise 2.3, Q1
Pairs ke har set ke liye batao ki woh function hai ya nahi; jo functions hain unka domain aur range batao.
1. ${(- 2, 4), (- 1, 1), (0, 0), (1, 1), (2, 4)}$
  Solution dikhao
  Function hai: koi pehli entry repeat nahi hoti. Domain ${- 2, - 1, 0, 1, 2}$ ; range ${0, 1, 4}$ .
2. ${(4, 2), (4, - 2), (9, 3), (9, - 3)}$
  Solution dikhao
  Function nahi hai: 4 ki do images 2 aur $- 2$ hain (aur 9 ki bhi do hain).
3. ${(5, 1), (6, 1), (7, 0)}$
  Solution dikhao
  Function hai. Domain ${5, 6, 7}$ ; range ${0, 1}$ .
2
Short answerMediumNCERT Exercise 2.3, Q2
Har real function ka domain aur range nikalo.
1. $f (x) = 3 - ∣ x ∣$
  Solution dikhao
  Har real $x$ allowed hai, isliye domain $𝐑$ hai. Kyunki $∣ x ∣ \geq 0$ , $f (x) \leq 3$ , aur jab $t = ∣ x ∣$ , $[0, \infty)$ par chalta hai to $f (x) = 3 - t$ har value $\leq 3$ leta hai. Range $(- \infty, 3]$ .
2. $f (x) = \sqrt{16 - x^{2}}$
  Solution dikhao
  $16 - x^{2} \geq 0$ chahiye, yaani $- 4 \leq x \leq 4$ : domain $[- 4, 4]$ . Phir $16 - x^{2}$ , 0 ( $x = \pm 4$ par) se 16 ( $x = 0$ par) tak jaata hai, isliye range $[0, 4]$ hai.
3
Short answerEasyNCERT Exercise 2.3, Q3
$g (x) = 7 - 3 x$ ke liye neeche di gayi values nikalo.
1. $g (0)$
  Answer dekho
  Answer 7
  $7 - 0 = 7$ .
2. $g (4)$
  Answer dekho
  Answer -5
  $7 - 12 = - 5$ .
3. $g (- 2)$
  Answer dekho
  Answer 13
  $7 + 6 = 13$ .
4
Short answerMediumNCERT Exercise 2.3, Q4
Function $s (F) = \frac{5 (F - 32)}{9}$ , $F$ degrees Fahrenheit ke temperature ko degrees Celsius mein badalta hai.
1. $s (212)$ nikalo.
  Answer dekho
  Answer 100
  $\frac{5 \times 180}{9} = 100$ : paani $212^{\circ} F$ par ubalta hai.
2. $s (98.6)$ nikalo.
  Answer dekho
  Answer 37
  $\frac{5 \times 66.6}{9} = 37$ : body ka normal temperature.
3. $s (- 40)$ nikalo.
  Answer dekho
  Answer -40
  $\frac{5 \times (- 72)}{9} = - 40$ : dono scales $- 40$ par match karte hain.
4. Jab $s (F) = 25$ ho to $F$ nikalo.
  Answer dekho
  Answer 77
  $\frac{5 (F - 32)}{9} = 25$ se $F - 32 = 45$ , isliye $F = 77$ .
5
Short answerMediumNCERT Exercise 2.3, Q5
Har function ki range nikalo.
1. $f (x) = 4 x + 1$ , $x \in 𝐑$ , $x < 0$
  Solution dikhao
  $x < 0$ se $4 x < 0$ , isliye $f (x) < 1$ ; har $y < 1$ , $x = \frac{y - 1}{4} < 0$ se aata hai. Range $(- \infty, 1)$ .
2. $f (x) = 3 - x^{2}$ , $x \in 𝐑$
  Solution dikhao
  $x^{2} \geq 0$ , isliye $f (x) \leq 3$ , aur $\leq 3$ wali har value milti hai. Range $(- \infty, 3]$ .
3. $f (x) = - x$ , $x \in 𝐑$
  Solution dikhao
  Har real $y$ , $f (- y)$ hai, isliye range $𝐑$ hai.

Miscellaneous Examples

Yeh examples chapter ke ideas ko jodte hain aur un methods ki jhalak dete hain jo tum coordinate geometry aur calculus mein phir use karoge.

Example 2.18NCERT Example 18 · p. 38

Linear function ka graph

$f : 𝐑 \to 𝐑$ , $f (x) = 6 - 2 x$ ka graph sketch karo, aur dekho ki yeh axes se kahan milta hai.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

2/2Ek step right, do steps neeche: slope $- 2$ , $y$ -intercept 6.

Fig. 2.14Linear function ka graph ek straight line hota hai.NCERT Fig. 2.16 · p. 38

Axes jinmein x minus 2 se 5 tak aur y minus 4 se 10 tak hai. Pehle step mein paanch dots (0, 6), (1, 4), (2, 2), (3, 0) aur (4, −2) mark karte hain, aur intercepts (0, 6) aur (3, 0) par labels hain. Doosre step mein ek straight line in sabse guzarti hai, aur (1, 4) se (2, 4) se (2, 2) tak ek dashed right-angled step dikhata hai ki ek unit right jaane ka matlab do units neeche; labels slope m = −2 aur y-intercept c = 6 batate hain.

Linear functions. Constants $m$ aur $c$ ke saath koi bhi function $f (x) = m x + c$ linear function kehlata hai. Iska graph slope $m$ wali ek straight line hai jo $y$ -axis ko $(0, c)$ par kaatti hai. Identity function ( $m = 1$ , $c = 0$ ) aur constant functions ( $m = 0$ ) iske special cases hain; Chapter 9 straight lines ko detail mein padhta hai.

Example 2.19NCERT Example 19 · p. 39

Integers par ek relation

Maan lo $R = {(a, b) : a, b \in 𝐙 and a + b is even}$ . Dikhao ki (i) har integer $a$ ke liye $(a, a) \in R$ ; (ii) agar $(a, b) \in R$ , to $(b, a) \in R$ ; (iii) agar $(a, b) \in R$ aur $(b, c) \in R$ , to $(a, c) \in R$ .

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.20NCERT Example 20 · p. 39

Linear rule dhoondhna

$𝐙$ par ek linear function $f (x) = m x + c$ values $f (- 2) = 7$ , $f (0) = 3$ , $f (1) = 1$ aur $f (3) = - 3$ leta hai. $f (x)$ nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.21NCERT Example 21 · p. 39

Rational function ka domain

$f (x) = \frac{2 x + 3}{x^{2} + x - 6}$ ka domain nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

Example 2.22NCERT Example 22 · p. 39

Do pieces mein ek function

Ek function $x < 1$ ke liye $f (x) = 2 - x$ aur $x \geq 1$ ke liye $f (x) = x + 1$ se diya gaya hai. Iska graph banao aur range nikalo.

Pehle khud try karo. Phir solution ek-ek step karke kholo.

3/3Line $x = 1$ graph se ek baar milti hai: $f (1) = 2$ .

Fig. 2.15Bhara dot graph ka hissa hai; hollow dot nahi.NCERT Fig. 2.17 · p. 39

Axes jinmein x minus 3 se 5 tak aur y minus 1 se 7 tak hai. Pehle step mein left piece, line y = 2 − x, (−3, 5) se (1, 1) ke hollow circle ki taraf girti hai. Doosre step mein right piece, line y = x + 1, (1, 2) ke bhare dot se (5, 6) tak uthti hai. Teesre step mein x = 1 par ek dashed vertical line banti hai aur bhare dot (1, 2) par ring lagti hai: line graph se sirf ek baar milti hai, isliye f(1) = 2 aur f ek function hai.

Quick check

3 mein se 1

Ek linear function ke liye

f (0) = - 4

aur

f (2) = 2

f (x)

kya hai?

2 x - 4

3 x - 4

- 4 x + 3

3 x + 2

Answer dekho

Answer (B) $3 x - 4$

f (0) = c = - 4

, phir

2 m - 4 = 2

m = 3

. Option A value 2 ko slope maan leta hai, lekin 6 ki rise 2 ke run mein hoti hai.

Recap

Summary aur key points

Ordered pair $(a, b)$ ki ek pehli aur ek doosri entry hoti hai, aur $(a, b) = (c, d)$ theek tab jab $a = c$ aur $b = d$ (Section 2.2).
$A \times B = {(a, b) : a \in A, b \in B}$ , jahan $n (A \times B) = n (A) \cdot n (B)$ aur $A \times \emptyset = \emptyset$ . Aam taur par $A \times B \neq B \times A$ .
$𝐑 \times 𝐑$ coordinate plane hai aur $𝐑 \times 𝐑 \times 𝐑$ space hai.
$A$ se $B$ tak ka relation $A \times B$ ka subset hai, jise roster form, set-builder form ya arrow diagram mein likhte hain (Section 2.3).
Domain pehli entries ka set hai, range images ka set hai, aur codomain poora $B$ hai; range codomain ka subset hai.
$p$ elements wale set se $q$ elements wale set tak $2^{p q}$ relations hote hain ((2.2)).
Function $f : A \to B$ aisa relation hai jismein $A$ ke har element ki theek ek image hoti hai; inputs ek image share kar sakte hain (Section 2.4).
Real function ka domain aur range $𝐑$ ke andar hote hain; formula ka domain reals ka woh sabse bada set hai jahan uska matlab banta hai.
Identity ( $y = x$ ), constant ( $y = c$ ), polynomial aur rational functions, saath mein parabola $x^{2}$ , cubic $x^{3}$ aur do branches wala $\frac{1}{x}$ (Section 2.4.1).
$∣ x ∣$ ek V hai jiski range $[0, \infty)$ hai, $sgn x$ ki range ${- 1, 0, 1}$ hai, aur $[x]$ ek staircase hai jiski range $𝐙$ hai.
$f \pm g$ , $f g$ , $α f$ aur $\frac{f}{g}$ pointwise define hote hain; $\frac{f}{g}$ ke liye $g (x) \neq 0$ chahiye (Section 2.4.2).
Linear function $m x + c$ ka graph slope $m$ wali straight line hai; piecewise function ko har break point par ek hi value chahiye (Miscellaneous Examples).

Timeline

Function ka idea kaise bana

1637
Descartes
La Géométrie curves ko coordinates ke pairs se describe karti hai, yahi Cartesian product ke peeche ka idea hai.
1673
Leibniz
Pehli baar shabd 'function' use karte hain, curve se judi quantities ke liye.
1718
Johann Bernoulli
Variable aur constants se bani kisi bhi quantity ko function kehte hain.
c. 1734
Euler
Notation f(x) shuru karte hain.
1748
Euler ki Introductio
Formulas ke roop mein functions ko analysis ka centre bana deti hai.
1837
Dirichlet
Koi bhi rule allow karte hain jo har x ko ek pakka y de, formula ho ya nahi.
1939
Bourbaki
Function ko ordered pairs ke set ki tarah define karte hain, yaani Cartesian product ka subset, jaise is chapter mein.

Apna NCERT exercise question dhoondo

Hamare practice questions ncert.ai ke liye likhe gaye hain aur NCERT exercises ko follow karte hain. Apni textbook ke question se match karne wala question dhoondo; uska answer uske saath hai.

Tumhari NCERT textbook mein	Yahan practice question
NCERT Exercise 2.1, Q1	Practice 2.1 · Q1
NCERT Exercise 2.1, Q2	Practice 2.1 · Q2
NCERT Exercise 2.1, Q3	Practice 2.1 · Q3
NCERT Exercise 2.1, Q4	Practice 2.1 · Q4
NCERT Exercise 2.1, Q5	Practice 2.1 · Q5
NCERT Exercise 2.1, Q6	Practice 2.1 · Q6
NCERT Exercise 2.1, Q7	Practice 2.1 · Q7
NCERT Exercise 2.1, Q8	Practice 2.1 · Q8
NCERT Exercise 2.1, Q9	Practice 2.1 · Q9
NCERT Exercise 2.1, Q10	Practice 2.1 · Q10
NCERT Exercise 2.2, Q1	Practice 2.2 · Q1
NCERT Exercise 2.2, Q2	Practice 2.2 · Q2
NCERT Exercise 2.2, Q3	Practice 2.2 · Q3
NCERT Exercise 2.2, Q4	Practice 2.2 · Q4
NCERT Exercise 2.2, Q5	Practice 2.2 · Q5
NCERT Exercise 2.2, Q6	Practice 2.2 · Q6
NCERT Exercise 2.2, Q7	Practice 2.2 · Q7
NCERT Exercise 2.2, Q8	Practice 2.2 · Q8
NCERT Exercise 2.2, Q9	Practice 2.2 · Q9
NCERT Exercise 2.3, Q1	Practice 2.3 · Q1
NCERT Exercise 2.3, Q2	Practice 2.3 · Q2
NCERT Exercise 2.3, Q3	Practice 2.3 · Q3
NCERT Exercise 2.3, Q4	Practice 2.3 · Q4
NCERT Exercise 2.3, Q5	Practice 2.3 · Q5
NCERT Miscellaneous Exercise, Q1	Miscellaneous exercises · Q1
NCERT Miscellaneous Exercise, Q2	Miscellaneous exercises · Q2
NCERT Miscellaneous Exercise, Q3	Miscellaneous exercises · Q3
NCERT Miscellaneous Exercise, Q4	Miscellaneous exercises · Q4
NCERT Miscellaneous Exercise, Q5	Miscellaneous exercises · Q5
NCERT Miscellaneous Exercise, Q6	Miscellaneous exercises · Q6
NCERT Miscellaneous Exercise, Q7	Miscellaneous exercises · Q7
NCERT Miscellaneous Exercise, Q8	Miscellaneous exercises · Q8
NCERT Miscellaneous Exercise, Q9	Miscellaneous exercises · Q9
NCERT Miscellaneous Exercise, Q10	Miscellaneous exercises · Q10
NCERT Miscellaneous Exercise, Q11	Miscellaneous exercises · Q11
NCERT Miscellaneous Exercise, Q12	Miscellaneous exercises · Q12

Quick check: MCQs

2.2 Sets ke Cartesian Products4 questions
2.3 Relations4 questions
2.4 Functions3 questions
2.4.1 Kuch functions aur unke graphs4 questions
2.4.2 Real functions ka algebra3 questions
Miscellaneous Examples3 questions

Glossary · 14

Key terms (English + Hindi)

Cartesian productकार्तीय गुणनkartiya gunan: $A \times B$ , un saare ordered pairs $(a, b)$ ka set jahan $a \in A$ aur $b \in B$ .Chapter mein pehli baar yahan aaya →
codomainसहप्रांतsahaprant: $A$ se $B$ tak ke relation ya function ke liye set $B$ .Chapter mein pehli baar yahan aaya →

domainप्रांतprant: Relation ki pehli entries ka set; function $f : A \to B$ ke liye poora $A$ .Chapter mein pehli baar yahan aaya →

functionफलनphalan: $A$ se $B$ tak ka aisa relation jismein $A$ ke har element ki theek ek image ho.Chapter mein pehli baar yahan aaya →

greatest integer functionमहत्तम पूर्णांक फलनmahattam purnank phalan: $[x]$ , $x$ se zyada na hone wala sabse bada integer; jaise $[2.7] = 2$ aur $[- 2.7] = - 3$ .Chapter mein pehli baar yahan aaya →

imageप्रतिबिंबpratibimb: Agar $(a, b)$ kisi relation mein hai, to $b$ , $a$ ki ek image hai; function ke liye $f (a)$ , $a$ ki image hai.Chapter mein pehli baar yahan aaya →

modulus functionमापांक फलनmapank phalan: $∣ x ∣$ , jo $x \geq 0$ ke liye $x$ aur $x < 0$ ke liye $- x$ ke barabar hai.Chapter mein pehli baar yahan aaya →

ordered pairक्रमित युग्मkramit yugm: Aisa pair $(a, b)$ jismein order matter karta hai: $(a, b) = (c, d)$ sirf tab jab $a = c$ aur $b = d$ .Chapter mein pehli baar yahan aaya →

polynomial functionबहुपद फलनbahupad phalan: $a_{0} + a_{1} x + \dots + a_{n} x^{n}$ , real coefficients aur whole-number powers ke saath.Chapter mein pehli baar yahan aaya →

rangeपरिसरparisar: Saari images ka set; yeh hamesha codomain ka subset hota hai.Chapter mein pehli baar yahan aaya →
rational functionपरिमेय फलनparimey phalan: Polynomial functions ka quotient $\frac{p (x)}{q (x)}$ , jo wahan defined hai jahan $q (x) \neq 0$ .Chapter mein pehli baar yahan aaya →
real functionवास्तविक फलनvastavik phalan: Aisa function jiska domain aur range $𝐑$ ke subsets hon.Chapter mein pehli baar yahan aaya →
relationसंबंधsambandh: $A \times B$ ka subset, jo aam taur par pehli aur doosri entries ko jodne wale rule se chuna jaata hai.Chapter mein pehli baar yahan aaya →

signum functionचिह्न फलनchihn phalan: $sgn x$ : $x > 0$ ke liye 1, $x = 0$ par 0, $x < 0$ ke liye $- 1$ .Chapter mein pehli baar yahan aaya →

Relations and Functions: Class 11 Maths Chapter 2 Hinglish mein

Is chapter mein aap kya seekhenge

2.1 Introduction

2.2 Sets ke Cartesian Products

Practice 2.1: answers ke saath

2.3 Relations

Practice 2.2: answers ke saath

2.4 Functions

2.4.1 Kuch functions aur unke graphs

2.4.2 Real functions ka algebra

Practice 2.3: answers ke saath

Miscellaneous Examples

Summary aur key points

Practice questions answers ke saath (NCERT “Miscellaneous exercises”)

Apna NCERT exercise question dhoondo

Quick check: MCQs

Key terms (English + Hindi)

Tum chapter ke end tak pahunch gaye