//M0//QN1//SUB//DL0//EQ

òu A = {1, 2, 3, 4, 5, 6}, R = {(x, y) : y = x + 1} ÚkkÞ íku heíku MktçktÄ R Au, su AÚke A Ãkh ÔÞkÏÞkrÞík Au, íkku (i) yk MktçktÄLku rfhý ykf]rík îkhk Ëþkoðku. (ii) RLkku «Ëuþ, Mkn«Ëuþ íku{s rðMíkkh {u¤ðku.

//X

ynª, x, y ∈ A = {1, 2, 3, 4, 5, 6} íkÚkk y = x + 1 Au.
x = 1, 2, 3, 4, 5 ÷uíkkt yLke ®f{íkku yLkw¢{u 2, 3, 4, 5, 6 {¤u. R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}
RLkku «Ëuþ : {1, 2, 3, 4, 5},
rðMíkkh : {2, 3, 4, 5, 6}
Mkn«Ëuþ : {1, 2, 3, 4, 5, 6}

//M0//QN2//SUB//DL0//EQ

Lke[uLke ykf]rík{kt PÚke QLkku MktçktÄ Ëþkoðu÷ Au. yk MktçktÄLku (i) økwýÄ{oLke heíku, (ii) ÞkËeLke heíku ÷¾ku. íkuLkku «Ëuþ yLku rðMíkkh þkuÄku.

//X

økwýÄ{oLke heík : R : {(x, y) : x = y 2, x ∈ P, y ∈ Q}
ÞkËeLke heík : R : {(9, 3), (9, –3), (4, 2), (4, –2),
(25, 5), (25, –5)}
«Ëuþ : {4, 9, 25}, rðMíkkh : {–2, 2, –3, 3, –5, 5}

//M0//QN3//SUB//DL0//EQ

òu A = {1, 2}, B = {3, 4}, íkku AÚke BLkk MktçktÄkuLke MktÏÞk þkuÄku.

//X

A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.
ynª, n(A) = 2, n(B) = 2 íkÚkk
n(A×B) = n(A) ⋅ n(B) = 2 × 2 = 4 {kxu MktçktÄkuLke MktÏÞk = 2 4 = 16 ÚkkÞ.

//M0//QN4//SUB//DL0

A = {1, 2, 3, 4, ..., 14}, R = {(x, y) : 3 x – y = 0,

x, y ∈ A}. òu R yu AÚke ALkku MktçktÄ nkuÞ, íkku RLkku «Ëuþ, Mkn«Ëuþ yLku rðMíkkh {u¤ðku.

//X

ynª, 3 x – y = 0, x, y ∈ {1, 2, 3, ..., 14}
\ 3 x = y
ynª, x = 1, 2, 3, 4 ÷uíkkt, yLke ®f{íkku yLkw¢{u
3, 6, 9, 12 {¤u.
\ R = {(1, 3), (2, 6), (3, 9), (4, 12)} {¤u.
\ R Lkku «Ëuþ : {1, 2, 3, 4}
(Ëhuf ¢{Þwõík òuzLkku «Úk{ ½xf)
\ R Lkku rðMíkkh : {3, 6, 9, 12}
(Ëhuf ¢{Þwõík òuzLkku çkeòu ½xf)
\ R Lkku Mkn«Ëuþ : øký A Ãkkuíku = {1, 2, 3, 4, ..., 14}

//M0//QN5//SUB//DL0

R = {(x, y) : y = x + 5, x yu 4Úke LkkLke «kf]ríkf MktÏÞk Au, x, y ∈ N} ÚkkÞ íku heíku yuf MktçktÄ N Ãkh ÔÞkÏÞkrÞík Au. RLku ÞkËeLke heíku ÷¾ku. RLkku «Ëuþ íku{s rðMíkkh {u¤ðku.

//X

ynª, x = 1, 2, 3 ÷uíkkt, yLke ®f{íkku
y = x + 5 ÃkhÚke yLkw¢{u 6, 7, 8 {¤u.
\ R = {(1, 6), (2, 7), (3, 8)}
\ R Lkku «Ëuþ : {1, 2, 3} íkÚkk R Lkku rðMíkkh = {6, 7, 8}

//M0//QN6//SUB//DL0

A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : x yLku yLkku íkVkðík yÞwø{ MktÏÞk Au. x ∈ A, y ∈ B} ÚkkÞ íku heíku MktçktÄ AÚke B Ãkh ÔÞkÏÞkrÞík Au. RLku ÞkËeLke heíku ÷¾ku.

//X

ynª, x ∈ A = {1, 2, 3, 5}, y ∈ B = {4, 6, 9} íkÚkk R : x yLku y Lkku íkVkðík yÞwø{ MktÏÞk Au.
R : {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}

//M0//QN7//SUB//DL0//EQ

Lke[uLke ykf]rík{kt PÚke QLkku MktçktÄ Ëþkoðu÷ Au. yk MktçktÄLku (i) økwýÄ{oLke heíku, (ii) ÞkËeLke heíku ÷¾ku. íkuLkku «Ëuþ yLku rðMíkkh þwt Úkþu ?

//X

P = {5, 6, 7}, Q = {3, 4, 5}
(i) økwýÄ{oLke heík : R = {(x, y) : x ∈ P, y ∈ Q,
x – y = 2}
(ii) ÞkËeLke heík : R = {(5, 3), (6, 4), (7, 5)}
RLkku «Ëuþ : {5, 6, 7},
RLkku rðMíkkh : {3, 4, 5}

//M0//QN8//SUB//DL0

òu A = {1, 2, 3, 4, 6}, R = {(a, b) : a, b ∈ A, b yu a ðzu rð¼kßÞ Au} íku heíku MktçktÄ R yu A Ãkh ÔÞkÏÞkrÞík Au. íkku RLku ÞkËeLke heíku ÷¾ku íkÚkk RLkku «Ëuþ yLku rðMíkkh {u¤ðku.

//X

ynª, a, b ∈ A = {1, 2, 3, 4, 6} íkÚkk R : b yu a ðzu rð¼kßÞ Au.

R = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2),
(2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}

«Ëuþ : {1, 2, 3, 4, 6}
rðMíkkh : {1, 2, 3, 4, 6}

//M0//QN9//SUB//DL0

R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}} ÚkkÞ íku heíku ÔÞkÏÞkrÞík MktçktÄLkku «Ëuþ íku{s rðMíkkh {u¤ðku.

//X

ynª, x ∈ {0, 1, 2, 3, 4, 5} íkÚkk (x, x + 5) {kxuLke ¢{Þwõík òuz R : {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} Au.

- \ RLkku «Ëuþ : {0, 1, 2, 3, 4, 5} íkÚkk

R Lkku rðMíkkh : {5, 6, 7, 8, 9, 10}

//M0//QN10//SUB//DL0

MktçktÄ R = {(x, x 3) : x yu 10 fhíkkt LkkLke yrð¼kßÞ MktÏÞk Au}Lku ÞkËeLkk MðYÃk{kt ÷¾ku.

//X

ynª, x = {2, 3, 5, 7} Au, {kxu (x, x 3) {kxuLke ¢{Þwõík òuzku R : {(2, 8), (3, 27), (5, 125), (7, 343)} ÚkkÞ.
R = {(2, 2 3), (3, 3 3), (5, 5 3), (7, 7 3)}

//M0//QN11//SUB//DL0

òu A = {x, y, z}; B = {1, 2}, íkku A Úke B Lkk MktçktÄkuLke MktÏÞk þkuÄku.

//X

ynª, n(A) = 3, n(B) = 2, íkku n(A × B) = n(A) ⋅ n(B) ÃkhÚke n(A × B) = 6 ÚkkÞ. íkuÚke AÚke BLkk MktçktÄkuLke MktÏÞk = 2 6 = 64 ÚkkÞ.

//M0//QN12//SUB//DL0//EQ

R yu Z Ãkh R = {(a, b) : a, b ∈ Z, a – b yu ÃkqýkOf Au} îkhk ÔÞkÏÞkrÞík Au. RLkku «Ëuþ yLku rðMíkkh þkuÄku.

//X

ynª, Z = {..., –2, –1, 0, 1, 2, ...} Au íkÚkk
(a, b) ∈ Z {kxu a – b yu ÃkqýkOf Au.
RLkku «Ëuþ yLku rðMíkkh çktLku Z Úkþu.
rðÄuÞ yu ÞtºkLke su{ Au, su yLkLÞ ykWxÃkwx Ëhuf RLkÃkwx {kxu ykÃku Au.
ÔÞkÏÞk : yrhõík øký A yLku B {kxu, MktçktÄ f îkhk øký ALkk «íÞuf ½xfLku Mktøkík øký B{kt yLkLÞ «rík®çkçk {¤u Au, íkku yk MktçktÄ fLku AÚke B ÃkhLkwt rðÄuÞ fnu Au. AÚke B ÃkhLkk rðÄuÞLku f : A → B ÷¾kÞ Au.
òu a ∈ A, f(a) = b, b ∈ B íkku bLku f îkhk {¤íkwt aLkwt «rík®çkçk fnuðkÞ Au. aLku f îkhk bLkwt Ãkqðo«rík®çkçk fnuðkÞ Au.
yk{, rðÄuÞ yu rðrþ»x «fkhLkku MktçktÄ Au.
fkuE MktçktÄ yu rðÄuÞ íÞkhu s çkLku ßÞkhu, «ËuþLkku Ëhuf ½xf Mkn«Ëuþ{ktLkk {kºk yLku {kºk yuf s ½xf MkkÚku òuzkÞu÷ nkuÞ yLÞÚkk ykÃku÷ MktçktÄ yu rðÄuÞ çkLku Lknª.
WËknhý íkhefu,
ynª ykf]rík (i){kt Ëþkoðu÷ MktçktÄ R 1 yu rðÄuÞ LkÚke, fkhý fu «Ëuþ{ktLkku ½xf 3 yu Mkn«ËuþLkk fkuE Ãký ½xf MkkÚku òuzkÞu÷ LkÚke.
ynª ykf]rík (ii){kt Ëþkoðu÷ MktçktÄ R 2 yu rðÄuÞ Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.
ynª ykf]rík (iii){kt Ëþkoðu÷ MktçktÄ R 3 yu rðÄuÞ LkÚke, fkhý fu «Úk{ ½xf 1 yu (Mkn«ËuþLkkt çku) ½xfku MkkÚku òuzkÞu÷ Au.
Äkhku fu, f : A → B nkuÞ, íkku øký ALku fLkku «Ëuþ fnu Au íkÚkk øký BLku fLkku Mkn«Ëuþ fnu Au íkÚkk ALkk çkÄk ½xfkuLkk «rík®çkçkLku fLkku rðMíkkh fnu Au.
ynª, MÃk»x Au fu rðMíkkh ⊆ Mkn«Ëuþ.
òu y-yûkLku Mk{ktíkh Ëkuhu÷ hu¾k ykÃku÷ ð¢Lku yuf fhíkkt ðÄkhu ®çkËwyku{kt AuËu íkku ð¢ yu Võík MktçktÄ Ëþkoðu Au, Ãkhtíkw rðÄuÞ Lk Ëþkoðu.
Ãkhtíkw òu y-yûkLku Mk{ktíkh Ëkuhu÷ hu¾k ykÃku÷ ð¢Lku yuf s ®çkËw{kt AuËu íkku ð¢ yu rðÄuÞ Ëþkoðu Au.
(1) íkËuð rðÄuÞ (Identity Function) :
f : R → R, ∀x ∈ R {kxu f(x) = x Lku íkËuð rðÄuÞ fnu Au
ykLkku yÚko yu ÚkkÞ fu, su RLkÃkwx nkuÞ íku s ykWxÃkwx {¤u.
ynª, íkËuð rðÄuÞLkku «Ëuþ yLku rðMíkkh R Au.
íkËuð rðÄuÞLkku yk÷u¾ Wøk{®çkËw{ktÚke ÃkMkkh Úkíke hu¾k Ëþkoðu Au.
(2) y[¤ rðÄuÞ (Constant Function) :
f : R → R, ∀x ∈ R, f(x) = c, (ßÞkt, c yu fkuE y[¤ Au)Lku y[¤ rðÄuÞ fnu Au.
y[¤ rðÄuÞLkku «Ëuþ R yLku íkuLkku rðMíkkh {c} Au.
y[¤ rðÄuÞLkku yk÷u¾ X-yûkLku Mk{ktíkh hu¾k Au.
òu c > 0, íkku hu¾k X-yûkLke WÃkh Au.
òu c = 0, íkku hu¾k X-yûk Ãkh MktÃkkíke nþu.
òu c < 0, íkku hu¾k X-yûkLke Lke[u Au.
(3) çknwÃkËe rðÄuÞ (Polynomial Function) :
f : R → R, ∀x ∈ R {kxu
f(x) = a 0 + a 1 x + a 2 x 2 + ... + a n x n, Lku n ½kíkLkwt çknwÃkËe rðÄuÞ fnu Au. ynª, n yu yLk]ý ÃkqýkOf Au.
a 0, a 1, a 2, ..., a n ∈ R íkÚkk a n ≠ 0.
WËk. f (x) = x 3 – x 2 + 2, g(x) = x 4 + x Au.
h(x) = + 2 x LkÚke.
(4) Mkt{uÞ rðÄuÞ (Rational Function) :
g(x) ≠ 0 nkuÞ íkuðk «Ëuþ{kt ÔÞkÏÞkrÞík çknwÃkËe rðÄuÞ f(x) yLku g(x) {kxu Lku Mkt{uÞ rðÄuÞ fnu Au.
(5) {kLkktf rðÄuÞ (Modulus Function) :
f(x) = |x| , f : R → RÚke ÔÞkÏÞkrÞík rðÄuÞLku {kLkktf rðÄuÞ fnu Au.
f(x) = |x| =
íkuðe s heíku,
|x – 1| = íkÚkk
|x + 2| = ÷¾e þfkÞ.
{kLkktf rðÄuÞLkku yk÷u¾ Y-yûk «íÞu Mktr{ík nkuÞ Au íkÚkk X-yûkLke WÃkh nkuÞ Au íkÚkk f(x) = |x|Lkku yk÷u¾ Wøk{®çkËw{ktÚke ÃkMkkh ÚkkÞ Au.
{kLkktf rðÄuÞLkku «Ëuþ R íkÚkk rðMíkkh [0, ∞) Au.
(6) r[nTLk rðÄuÞ (Signum Function) :
f : R → R {kxu, f(x) =
ðzu ÔÞkÏÞkrÞík rðÄuÞLku r[nTLk rðÄuÞ fnu Au.
«Ëuþ R yLku rðMíkkh {–1, 0, 1} Au.
r[nTLk rðÄuÞ f(x) = ðzu Ãký ËþkoðkÞ Au.
(7) {n¥k{ ÃkqýkOf rðÄuÞ (Greatest Integer Function) :
rðÄuÞ f : R → R, f(x) = [x] yÚkðk x,
∀x ∈ R Lku {n¥k{ ÃkqýkOf rðÄuÞ fnu Auu. ßÞkt, [x] yÚkðk
x yu xÚke LkkLkk yÚkðk xLku Mk{kLk nkuÞ íkuðk ík{k{ ÃkqýkOfku{kt MkkiÚke {kuxku ÃkqýkOf Ëþkoðu Au. íkuLku floor function íkhefu Ãký yku¤¾ðk{kt ykðu Au.
{n¥k{ ÃkqýkOf rðÄuÞLkku «Ëuþ R íkÚkk rðMíkkh Z Au.
fkuE Ãký x ∈ R Lkku {n¥k{ ÃkqýkOf Lke[u {wsçk þkuÄe þfkÞ Au :
[2.3] = 2, [– 3.2] = – 4, [5.99] = 5
[3] = 3, [0] = 0
fkuE Ãký ÃkqýkOf MktÏÞkLke {n¥k{ ÃkqýkOf íku MktÏÞk Ãkkuíku s ÚkkÞ.
(8) rMk®÷øk rðÄuÞ (Ceiling Function) :
rðÄuÞ f : R → R, f(x) = x Lku rMk®÷øk rðÄuÞ fnu Au.
rMk®÷øk rðÄuÞLkku «Ëuþ R yLku rðMíkkh ‘Z’ Au.
fkuE Ãký x ∈ R Lkku rMk®÷øk rðÄuÞ Lke[u {wsçk þkuÄe þfkÞ Au :
2.3 = 3, –3.2 = – 4, 5.99 = 5
3 = 3, 0 = 0
fkuE Ãký ÃkqýkOf MktÏÞkLkku rMk®÷øk rðÄuÞ íku MktÏÞk Ãkkuíku s ÚkkÞ.
(9) ÃkqýkOf ¼køk rðÄuÞ (Fractional Part function) :
rðÄuÞ f : R → R, f(x) = {x}
= x – [x]
Lku ÃkqýkOf ¼køk rðÄuÞ fnu Au.
ÃkqýkOf ¼køk rðÄuÞLkku R yLku rðMíkkh [0, 1) Au.
fkuE Ãký x ∈ R Lkku ÃkqýkOf ¼køk rðÄuÞ Lke[u {wsçk þkuÄe þfkÞ :
{2.3} = 0.3, {– 3.2} = 0.8
{5.99} = 0.99, {3} = 0
fkuE Ãký ÃkqýkOf MktÏÞkLkku ÃkqýkOf ¼køk rðÄuÞLkwt {qÕÞ ‘þqLÞ’ {¤u.
Äkhku fu, f : D 1 → R íkÚkk g : D 2 → R çku ðkMíkrðf rðÄuÞku Au.

(i) (f + g) (x) = f(x) + g(x), ∀x ∈ D 1 ∩ D 2

(ii) (f – g) (x) = f(x) – g(x), ∀x ∈ D 1 ∩ D 2

(iii) (fg) (x) = f(x) ⋅ g(x), ∀x ∈ D 1 ∩ D 2

(iv) (x) = ,

∀x ∈ (D 1 ∩ D 2) – {x : g(x) = 0}

(v) ðkMíkrðf rðÄuÞLkku y[¤ MkkÚku økwýkfkh : òu c yu fkuE y[¤ nkuÞ, íkku (c ⋅ f) (x) = c ⋅ f(x) ÚkkÞ.

//M0//QN13//SUB//DL0

N yu «kf]ríkf MktÏÞkykuLkku øký Au yLku íkuLke Ãkh ÔÞkÏÞkrÞík fkuE MktçktÄ R yuðku Au fu, R = {(x, y) :

y = 2 x, x, y ∈ N}, íkku RLkku «Ëuþ, Mkn«Ëuþ yLku rðMíkkh þkuÄku. þwt yk MktçktÄ rðÄuÞ Au ?

//X

ynª, x, y ∈ N íkÚkk y = 2 x Au. {kxu ynª, «Ëuþ N, Mkn«Ëuþ N íkÚkk rðMíkkh yu Þwø{ «kf]ríkf MktÏÞk Úkþu.
ynª, «íÞuf x ∈ N {kxu yLkLÞ y ∈ N {¤u Au, {kxu ykÃku÷ MktçktÄ yu rðÄuÞ çkLkþu.

//M0//QN14//SUB//DL0//EQ

Lke[uLkkt WËknhýku{kt ykÃku÷ MktçktÄ [fkMkku yLku «íÞuf MktçktÄ rðÄuÞ Au fu Lknª íku fkhý ykÃke sýkðku. (i) R = {(2, 1), (3, 1), (4, 2)} (ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)} (iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

//X

(i) R = {(2, 1), (3, 1), (4, 2)}Lku Lke[u {wsçk Ëþkoðe þfkÞ :

2

3

4

R

1

2

ynª, «Ëuþ{ktLkku Ëhuf ½xf Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ çkLku.
(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}

2

3

4

R

2

3

4

ynª, «Ëuþ{ktLkku ½xf 2 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ çkLku Lknª.
(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}
ynª, «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ Au.

//M0//QN15//SUB//DL0

N yu «kf]ríkf MktÏÞkykuLkku øký Au f : N → N, f(x) = 2 x + 1 îkhk ÔÞkÏÞkrÞík ðkMíkrðf rðÄuÞ Au. yk ÔÞkÏÞkLke {ËËÚke Lke[uLkwt fku»xf Ãkqýo fhku :

x | 1 | 2 | 3 | 4 | 5 | 6 | 7
y | f(1)=... | f(2)=... | f(3)=... | f(4)=... | f(5)=... | f(6)=... | f(7)=...

//X

ynª, f(x) = 2 x + 1 Au.
f(1) = 2(1) + 1 = 3, f(2) = 2(2) + 1 = 5, f(3) = 2(3) + 1 = 7
f(4) = 2(4) + 1 = 9, f(5) = 2(5) + 1 = 11, f(6) = 2(6) + 1 = 13,
f(7) = 2(7) + 1 = 15

x | 1 | 2 | 3 | 4 | 5 | 6 | 7
y | f(1)=3 | f(2)=5 | f(3)=7 | f(4)=9 | f(5)=11 | f(6)=13 | f(7)=15

//M0//QN16//SUB//DL0//EQ

f : R → R, y = f(x) = x 2, x ∈ R Úke ÔÞkÏÞkrÞík yuf rðÄuÞ Au. yk ÔÞkÏÞkLku ykÄkhu Lke[uLkwt fku»xf Ãkqýo fhku. yk rðÄuÞLkku «Ëuþ yLku rðMíkkh þwt Úkþu ? fLkku yk÷u¾ Ëkuhku.

x | – 4 | – 3 | – 2 | – 1 | 0 | 1 | 2 | 3 | 4
y=f(x)=x 2 |  |  |  |  |  |  |  |  |

//X

ynª, y = f(x) = x 2 Au.
\ f(– 4) = (– 4)2 = 16, f(– 3) = (– 3)2 = 9,
f(– 2) = (– 2)2 = 4, f(– 1) = (– 1)2 = 1,
f(0) = (0)2 = 0, f(1) = (1)2 = 1, f(2) = (2)2 = 4,
f(3) = (3)2 = 9, f(4) = (4)2 = 16 ÚkkÞ.

x | – 4 | – 3 | – 2 | – 1 | 0 | 1 | 2 | 3 | 4
y=f(x)=x 2 | 16 | 9 | 4 | 1 | 0 | 1 | 4 | 9 | 16

ynª, fLkku «Ëuþ R íkÚkk rðMíkkh [0, ∞) Úkþu.
y = f(x) = x 2Lkku yk÷u¾ Lke[u {wsçk Au: su Ãkhð÷Þ Ëþkoðu Au.

//M0//QN17//SUB//DL0//EQ

f : R → R, f(x) = x 3, x ∈ RÚke ÔÞkÏÞkrÞík rðÄuÞLkku yk÷u¾ Ëkuhku.

//X

f(x) = x 3 {kxu,
f(0) = (0)3 = 0, f(1) = (1)3 = 1,
f(– 1) = (– 1)3 = – 1, f(2) = (2)3 = 8,
f(– 2) = (– 2)3 = – 8, f(–3) = –27
f(3) = 27 ðøkuhu ...

//M0//QN18//SUB//DL0//EQ

f : R – {0} → R, f(x) = , x ∈ R – {0}Úke ÔÞkÏÞkrÞík yuf rðÄuÞ ykÃku÷ Au. yk ÔÞkÏÞkLkk ykÄkhu Lke[uLkwt fku»xf Ãkqýo fhku. yk rðÄuÞLkku «Ëuþ yLku rðMíkkh þwt Úkþu ?

x | – 2 | – 1.5 | – 1 | – 0.5 | 0.25 | 0.5 | 1 | 1.5 | 2
y = f(x) = | – | – | – | – | – | – | – | – | –

//X

y = f(x) = ykÃku÷ Au.

x | – 2 | – 1.5 | – 1 | – 0.5 | 0.25 | 0.5 | 1 | 1.5 | 2
y = | – 0.5 | –0.67 | – 1 | – 2 | 4 | 2 | 1 | 0.67 | 0.5

fk[ku yk÷u¾ :

//M0//QN19//SUB//DL0//EQ

f(x) = x 2, g(x) = 2 x + 1 yu çku ðkMíkrðf rðÄuÞku nkuÞ, íkku (f + g)(x), (f – g)(x), (fg)(x), (x) þkuÄku.

//X

(f + g)(x) = f(x) + g(x) = x 2 + 2 x + 1
(f – g)(x) = f(x) – g(x) = x 2 – (2 x + 1) = x 2 – 2 x – 1
(f ⋅ g)(x) = f(x) ⋅ g(x) = x 2 (2 x + 1) = 2 x 3 + x 2
(x) = , g(x) ≠ 0, = , x ≠

//M0//QN20//SUB//DL0//EQ

f(x) = , g(x) = x yu çku yLk]ý ðkMíkrðf MktÏÞkLkk øký Ãkh ÔÞkÏÞkrÞík rðÄuÞ nkuÞ, íkku (f + g)(x), (f – g) (x), (f ⋅ g)(x), (x) þkuÄku.

//X

(f + g)(x) = f(x) + g(x) = + x
(f – g)(x) = f(x) – g(x) = – x
(f ⋅ g)(x) = f(x) ⋅ g(x) = ⋅ x = x ⋅ x = x+1 = x
(x) = , g(x) ≠ 0, = , x ≠ 0,
= x–1 = x–, x ≠ 0

//M0//QN21//SUB//DL0//EQ

Lke[uLkk Ãkife fÞku MktçktÄ rðÄuÞ Au ? fkhý ykÃkku. òu íku rðÄuÞ nkuÞ, íkku íkuLkku «Ëuþ yLku rðMíkkh þkuÄku.(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

//X

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ Ëþkoðu Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.
«Ëuþ : {2, 5, 8, 11, 14, 17}
rðMíkkh : {1}
(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
ynª, ykÃku÷ MktçktÄ yu rðÄuÞ Ëþkoðu Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.
«Ëuþ : {2, 4, 6, 8, 10, 12, 14}
rðMíkkh : {1, 2, 3, 4, 5, 6, 7}
(iii) {(1, 3), (1, 5), (2, 5)}
ynª, ykÃku÷ MktçktÄ yu rðÄuÞ LkÚke, fkhý fu «Ëuþ{ktLkku ½xf 1 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷ Au.

//M0//QN22//SUB//DL0//EQ

Lke[uLkkt ðkMíkrðf rðÄuÞkuLkk «Ëuþ yLku rðMíkkh þkuÄku.(i) f(x) = – |x|

//X

ynª, f(x) = – |x|{kt xLke fkuE Ãký ðkMíkrðf ®f{ík {qfe þfkíke nkuðkÚke «Ëuþ R Úkþu.
íkÚkk |x|Lkku rðMíkkh [0, ∞) nkuðkÚke – |x|Lkku rðMíkkh
(– ∞, 0] ÚkkÞ.
(ii) f(x) =
«Ëuþ {u¤ððk {kxu,
\ 9 – x 2 ≥ 0
\ 9 > x 2
\ x 2 < 9
\ < 3
\ |x| < 3
\ – 3 < x < 3
\ «Ëuþ : x ∈ [– 3, 3]
rðMíkkh {u¤ððk {kxu,
x ∈ [– 3, 3] ⇒ – 3 < x < 3
⇒ 0 < x 2 < 9
⇒ – 9 < – x 2 < 0
⇒ 0 < 9 – x 2 < 9
⇒ 0 < < 3
⇒ 0 < f(x) < 3
\ f(x) ∈ [0, 3]
\ f(x) = Lkku rðMíkkh [0, 3] Au.

//M0//QN23//SUB//DL0

f(x) = 2 x – 5Úke ÔÞkÏÞkrÞík rðÄuÞ {kxu Lke[uLke ®f{íkku þkuÄku.(i) f(0)

//X

2(0) – 5 = 0 – 5 = – 5
(ii) f(7)
2(7) – 5 = 14 – 5 = 9
(iii) f(– 3)
2(– 3) – 5 = – 6 – 5 = – 11

//M0//QN24//SUB//DL0//EQ

rðÄuÞ ‘t’ yu MkuÂÕMkÞMk{kt W»ýíkk{kLk yLku VuhLknex{kt W»ýíkk{kLk ðå[u YÃkktíkh fhíkwt Mkqºk, t(c) = + 32 îkhk ÔÞkÏÞkrÞík Au, íkku Lke[uLkkt {qÕÞku þkuÄku.

//X

t(c) = + 32
(i) t(0)
(0) + 32 = 0 + 32 = 0
(ii) t(28)
(28) + 32 = + 32 = =
(iii) t(– 10)
(– 10) + 32 = –18 + 32 = 14
(iv) òu t(c) = 212 nkuÞ, íkku c þkuÄku.
\ 212 = c + 32
\ 212 – 32 = c
\ = c
\ c = 100

//M0//QN25//SUB//DL0

Lke[uLkk rðÄuÞkuLkk rðMíkkh þkuÄku.(i) f(x) = 2 – 3 x, x ∈ R, x > 0

//X

heík 1 :
ynª, f(x) = 2 – 3 x{kt x > 0Lke y÷øk y÷øk ®f{íkku {qfíkkt,

x | 0.01 | 0.1 | 0.9 | 1 | 2 | 2.5 | 4 | 5 | ...ðøkuhu
f(x) = | 1.97 | 1.7 | – 0.7 | – 1 | – 4 | – 5.5 | – 10 | – 13 | ...ðøkuhu

ynª, fku»xf ÃkhÚke òuE þfkÞ Au fu, ∀x > 0 {kxu,
f(x) < 2 \ rðMíkkh : (– ∞, 2) Úkþu.
heík 2 :
ynª, x > 0
\ 3 x > 0
\ – 3 x < 0
\ 2 – 3 x < 0 + 2
\ 2 – 3 x < 2
\ 2 – 3 x ∈ (– ∞, 2)
\ f(x) ∈ (– ∞, 2)
(ii) f(x) = x 2 + 2, x ∈ R
ynª, x 2 ≥ 0
\ x 2 + 2 ≥ 0 + 2
\ x 2 + 2 ≥ 2
\ rðMíkkh : [2, ∞) Úkþu.
(iii) f(x) = x, x ∈ R
ynª, f(x) = x yu íkËuð rðÄuÞ nkuðkÚke íkuLkku rðMíkkh R ÚkkÞ.

//M0//QN26//SUB//DL0//EQ

ðkMíkrðf MktÏÞk øký R Ãkh ÔÞkÏÞkrÞík ðkMíkrðf rðÄuÞ f : R → R, f(x) = x + 10, íkku rðÄuÞ fLkku yk÷u¾ Ëkuhku.

//X

f(x) = x + 10, x ∈ R ykÃku÷ Au. f(0) = 0 + 10 = 10,
f(1) = 1 + 10 = 11, f(–5) = –5 + 10 = 5, f(5) = 5 + 10 = 15 ðøkuhu ... ÷uíkkt yk÷u¾ Lke[u {wsçk {¤u :

//M0//QN27//SUB//DL0

òu R yu QÚke Q ÃkhLkku R = {(a, b) : a, b ∈ Q yLku a – b ∈ Z} ÚkkÞ íku heíku ÔÞkÏÞkrÞík MktçktÄ Au, íkku çkíkkðku fu, (i) «íÞuf a ∈ Q {kxu (a, a) ∈ R (ii) òu (a, b) ∈ R, íkku (b, a) ∈ R (iii) òu (a, b) ∈ R yLku (b, c) ∈ R, íkku (a, c) ∈ R.

//X

(i) Äkhku fu, a ∈ Q ÷uíkkt, {kxu a – a = 0 ∈ Z. íkuÚke (a, a) ∈ R ÚkkÞ.
(ii) òu (a, b) ∈ R, íkku a – b ∈ Z. íkuÚke b – a ∈ Z. íkuÚke (b, a) ∈ R.
(iii) òu (a, b) ∈ R yLku (b, c) ∈ R, íkku a – b ∈ Z.
b – c ∈ Z. íkuÚke a – c = (a – b) + (b – c) ∈ Z. íkuÚke (a, c) ∈ R ÚkkÞ.

//M0//QN28//SUB//DL0//EQ

f = {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} ÚkkÞ íku heíku

Z Ãkh ÔÞkÏÞkrÞík Mkwhu¾ rðÄuÞ Au, íkku f(x) þkuÄku.

//X

Äkhku fu, Mkwhu¾ rðÄuÞ f(x) = mx + c ÷uíkkt,
f(x) = mx + c íkÚkk f(1) = 1 f(2) = 3
\ f(1) = m(1) + c \ 3 = m(2) + c
\ 1 = m + c ...(1) \ 3 = 2 m + c ...(2)
Ãkrhýk{ (1) yLku (2) ÃkhÚke,
m = 2Lku (1){kt {qfíkkt,
\ m + c = 1
\ c = 1 – 2
\ c = – 1
\ f(x) = mx + c ÃkhÚke,
f(x) = 2 x – 1 ÚkkÞ.

//M0//QN29//SUB//DL0//EQ

f(x) = nkuÞ, íkku rðÄuÞLkku «Ëuþ þkuÄku.

//X

ykÃku÷ rðÄuÞ f yu Mkt{uÞ rðÄuÞ Au. {kxu fkuE Ãký Mkt{uÞ rðÄuÞLkku «Ëuþ = R – {x | AuË = 0},
ynª, x 2 – 5 x + 4 = 0 ÷uíkkt,
\ (x – 4) (x – 1) = 0
\ x = 4, x = 1
\ «Ëuþ : R – {1, 4}

//M0//QN30//SUB//DL0//EQ

f(x) = Úke ÔÞkÏÞkrÞík rðÄuÞLkku yk÷u¾ Ëkuhku.

//X

x = 0, íkku f(0) = 1 Au.
x = – 1,
íkku f(– 1) = 1 – (– 1) = 2, x = 2,
íkku f(2) = 2 + 1 = 3, x = 3,
íkku f(3) = 3 + 1 = 4, x = – 2,
íkku f(– 2) = 1 – (– 2) = 3 ÚkkÞ.

//M0//QN31//SUB//DL0//EQ

MktçktÄ f yu f(x) = Úke ÔÞkÏÞkrÞík Au yLku MktçktÄ g yu g(x) = Úke ÔÞkÏÞkrÞík Au, íkku Mkkrçkík fhku fu, f yu rðÄuÞ Au yLku g yu rðÄuÞ LkÚke.

//X

ynª, f(x) = ÃkhÚke,
ynª, ykf]rík ÃkhÚke òuE þfkÞ Au fu, «ËuþLkku Ëhuf ½xf Mkn«ËuþLkk yLkLÞ ({kºk yuf s) ½xf MkkÚku òuzkÞu÷ Au.
{kxu f yu rðÄuÞ Au.
g(x) =
ynª, ykf]rík ÃkhÚke òuE þfkÞ Au fu, «ËuþLkku ½xf 2 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷k nkuðkÚke g yu rðÄuÞ LkÚke.
x 2 = (2)2 = 4 = g(2)
3(2) = 6 = g(2)

//M0//QN32//SUB//DL0//EQ

f(x) = x 2, íkku þkuÄku.

//X

f(1.1) = (1.1)2 = 1.21, f(1) = (1)2 = 1
\ =
=
=
= = 2.1

//M0//QN33//SUB//DL0//EQ

rðÄuÞ f(x) = Lkku «Ëuþ þkuÄku.

//X

f(x) =
«Ëuþ {u¤ððk {kxu,
AuË ≠ 0
\ x 2 – 8 x + 12 ≠ 0
\ (x – 6) (x – 2) ≠ 0
\ x – 6 ≠ 0 \ x – 2 ≠ 0
\ x ≠ 6 \ x ≠ 2
\ x ∈ R – {2, 6}
\ «Ëuþ = R – {2, 6}

//M0//QN34//SUB//DL0//EQ

f(x) = Úke ÔÞkÏÞkrÞík rðÄuÞLkku «Ëuþ yLku rðMíkkh þkuÄku.

//X

f(x) =
«Ëuþ {u¤ððk {kxu,
x – 1 > 0
\ x > 1
\ x ∈ [1, ∞)
\ «Ëuþ : [1, ∞)
rðMíkkh {u¤ððk {kxu,
x ∈ [1, ∞) ⇒ x > 1
⇒ x – 1 > 0
⇒ > 0
⇒ f(x) > 0
⇒ f(x) ∈ [0, ∞)
\ f Lkku rðMíkkh = [0, ∞)

//M0//QN35//SUB//DL0

f(x) = |x – 1|Úke ÔÞkÏÞkrÞík rðÄuÞLkku «Ëuþ yLku rðMíkkh þkuÄku.

//X

x ∈ R ⇒ x – 1 ∈ R
\ f(x) Lkku «Ëuþ R Au.
x – 1 ∈ R ⇒ |x – 1| > 0
⇒ f(x) > 0
⇒ f(x) ∈ [0, ∞)
\ f(x)Lkku rðMíkkh = [0, ∞) Au.

//M0//QN36//SUB//DL0//EQ

òu f = yu RÚke RLkwt rðÄuÞ nkuÞ, íkku fLkku rðMíkkh þkuÄku.

//X

f =
ynª, x ∈ R ⇒ x 2 > 0
⇒ 1 + x 2 > 1
⇒ > 0
nðu, x 2 < x 2 + 1
\ < 1
\ 0 < < 1
\ 0 < f(x) < 1
\ f(x) ∈ [0, 1)
\ f(x)Lkku rðMíkkh [0, 1) Au.

//M0//QN37//SUB//DL0//EQ

f, g : R → R, f(x) = x + 1, g(x) = 2 x – 3Úke ÔÞkÏÞkrÞík rðÄuÞ Au, íkku f + g, f – g, þkuÄku.

//X

(f + g) (x) = f(x) + g(x) = x + 1 + 2 x – 3 = 3 x – 2
(f – g) (x) = f(x) – g(x) = x + 1 – (2 x – 3)
= x + 1 – 2 x + 3
= 4 – x
(x) = , g(x) ≠ 0, = , x ≠

//M0//QN38//SUB//DL0//EQ

òu f : {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} yu f(x) = ax + bÚke ÔÞkÏÞkrÞík MktçktÄ nkuÞ, íkku a yLku b þkuÄku.

//X

(1, 1) : f(1) = 1 ÃkhÚke,
f(x) = ax + b
\ f(1) = a(1) + b
\ 1 = a + b ...(1)
(2, 3) : f(2) = 3 ÃkhÚke,
f(x) = ax + b
\ f(2) = a(2) + b
\ 3 = 2 a + b ...(2)
(1) yLku (2) ÃkhÚke,
a = 2Lku Mk{e. (1){kt {qfíkkt,
\ a + b = 1
\ 2 + b = 1

//M0//QN39//SUB//DL0

R yu NÚke NLkku MktçktÄ Au. R : {(a, b) : a, b ∈ N yLku a = b 2} ÚkkÞ íku heíku ÔÞkÏÞkrÞík Au, íkku þwt Lke[uLkkt rðÄkLkku MkíÞ Au ? (i) ∀a ∈ N {kxu (a, a) ∈ R (ii) òu (a, b) ∈ R, íkku (b, a) ∈ R (iii) òu (a, b) ∈ R, (b, c) ∈ R, íkku (a, c) ∈ R «íÞuf rðÄkLk{kt ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

//X

ynª, R = {(a, b) : a, b ∈ N, a = b 2} ykÃku÷ Au.

(i) Äkhku fu, 3 ∈ N {kxu, 3 = (3)2 = 9 su þõÞ LkÚke, {kxu ∀a ∈ N {kxu (a, a) ∈ R MkíÞ LkÚke.

(ii) a = 9, b = 3 ÷uíkkt, (9, 3) ∈ R, fkhý fu 3 2 = 9 Ãkhtíkw (3, 9) ∉ R, fkhý fu 9 2 ≠ 3

\ ykÃku÷ rðÄkLk MkíÞ LkÚke.

(iii) a = 16, b = 4 ÷uíkkt, 4 2 = 16 ÚkkÞ.

\ (16, 4) ∈ R ÚkkÞ.

b = 4, c = 2 ÷uíkkt, 2 2 = 4 ÚkkÞ.

\ (4, 2) ∈ R Ãkhtíkw

(a, c) = (16, 2), ∉ R, fkhý fu 2 2 ≠ 16

\ ykÃku÷ rðÄkLk MkíÞ LkÚke.

//M0//QN40//SUB//DL0//EQ

A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} yLku

f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}, íkku þwt Lke[uLkkt rðÄkLkku MkíÞ Au ? (i) f yu AÚke B ÃkhLkku MktçktÄ Au. (ii) f yu AÚke B ÃkhLkwt rðÄuÞ Au. ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

//X

ynª, f yu A × BLkku WÃkøký nkuðkÚke f yu MktçktÄ Au, {kxu rðÄkLk (i) MkíÞ Au.
íkÚkk ykf]rík 5hÚke òuE þfkÞ Au fu, «ËuþLkku ½xf ‘2’ yu Mkn«ËuþLkk çku ½xfku MkkÚku òuzkÞu÷ Au, {kxu f yu rðÄuÞ LkÚke. {kxu rðÄkLk (ii) MkíÞ LkÚke.

//M0//QN41//SUB//DL0

f yu Z × ZLkku WÃkøký Au. òu f = {ab, a + b) : a, b ∈ Z} Úke ÔÞkÏÞkrÞík Au, íkku þwt f yu ZÚke ZLkwt rðÄuÞ Au ? ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

//X

f = {ab, a + b}, a, b ∈ Z
\ f (ab) = a + b
a = 1, b = 2 ÷uíkkt, f(1 × 2) = 1 + 2 \ f(2) = 3 {¤u.
a = –1, b = –2 ÷uíkkt, f(–1 × –2) = – 1 – 2
\ f(2) = – 3 {¤u.
yk{, (2, 3) yLku (2, – 3) çku ¢{Þwõík òuz {¤u.
yk{, 2 yu çku ½xfku MkkÚku òuzkÞu÷ Au.
\ f yu Z Úke Z Lkwt rðÄuÞ LkÚke.

//M0//QN42//SUB//DL0//EQ

A = {9, 10, 11, 12, 13}, f : A → N, f(n) = nLkku {n¥k{ yrð¼kßÞ yðÞð Au. fLkku rðMíkkh þkuÄku.

//X

f = A → N íkÚkk f(n) = nLkku {n¥k{ yrð¼kßÞ yðÞð
f(9) = 9Lkku {n¥k{ yrð¼kßÞ yðÞð = 3 íkuðe s heíku, f(10) = 5, f (9) = 3, f(11) = 11, f(12) = 3, f(13) = 13 {¤u.
\ fLkku rðMíkkh = {3, 5, 11, 13} ÚkkÞ.
2
«fhý Mkqr[
yøkkWLkk ð»ko{kt ÃkwAkÞu÷k JEE-Main «&LkkuLkwt «fhýðkh rð&÷u»ký