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![maths - Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_56656_303e7e3b_1772347754829_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=829_sc.jpg "maths - Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) - ShareChat")
![maths - Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X"-1 f(x) = Ixll "(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X"-1 f(x) = Ixll "(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X - ShareChat](https://cdn4.sharechat.com/33d5318_1c8/3174ddb4_1772289770826_sc.jpeg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=26_sc.jpeg "maths - Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X\"-1 f(x) = Ixll \"(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X\"-1 f(x) = Ixll \"(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X - ShareChat")
![biology - PHYUM PRIEERD They ஈீD $ cewh catel Qute goneollப IoAnQ ೧ 8pongiLla )_ ( للط لللمم اعرم mukceiuLor organuims org syslom_৮ prexnk wbld ConcQ Ilcten tronspont 01 hop cion /and excretion o @huton r೮೦ i೧ Digestioo fs lotroce tlulo ع spongio,Gibres_or-Spiculea & Skelekool mmode ob not sepecate ) Cexe Hemaphrodiel QL FeriuboD &sA(oterocQ develanh preLent) ( LccUcL_Stcge is Ind(roue Syoon , Spongillc bcH Spoge ೭೦ PHVUN CDELENTEBBTA CCNIDBRIB moskut swlDomio Iheg ore {oe sessyel moul ue ) 0 Mulieuloک okaorisms levell w &AuQ 09` Aada[U The ore diploblooticzand Slmmak?t onidoudtoz Cnioob@b pue೨enh ٥ oD kotack Cod9 and |e Cnfoblasbs /e_uoed_kur_ccboraq-e_sdebente] 0 Q0d Ihe ophura n pr 8 Dioetioo intrateuulcc & extq ckiulaC , and howe LaLm CrboDote CeaAl SRelet Cniclariq xbubit மooy bmப ೧ bolyp abd meduAQ _ Polyps & %LQ ZrdcQ bom_LRe ODd hudro ) adamstql PHYUM PRIEERD They ஈீD $ cewh catel Qute goneollப IoAnQ ೧ 8pongiLla )_ ( للط لللمم اعرم mukceiuLor organuims org syslom_৮ prexnk wbld ConcQ Ilcten tronspont 01 hop cion /and excretion o @huton r೮೦ i೧ Digestioo fs lotroce tlulo ع spongio,Gibres_or-Spiculea & Skelekool mmode ob not sepecate ) Cexe Hemaphrodiel QL FeriuboD &sA(oterocQ develanh preLent) ( LccUcL_Stcge is Ind(roue Syoon , Spongillc bcH Spoge ೭೦ PHVUN CDELENTEBBTA CCNIDBRIB moskut swlDomio Iheg ore {oe sessyel moul ue ) 0 Mulieuloک okaorisms levell w &AuQ 09` Aada[U The ore diploblooticzand Slmmak?t onidoudtoz Cnioob@b pue೨enh ٥ oD kotack Cod9 and |e Cnfoblasbs /e_uoed_kur_ccboraq-e_sdebente] 0 Q0d Ihe ophura n pr 8 Dioetioo intrateuulcc & extq ckiulaC , and howe LaLm CrboDote CeaAl SRelet Cniclariq xbubit மooy bmப ೧ bolyp abd meduAQ _ Polyps & %LQ ZrdcQ bom_LRe ODd hudro ) adamstql - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_227536_1f4fe84f_1772261226310_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=310_sc.jpg "biology - PHYUM PRIEERD They ஈீD $ cewh catel Qute goneollப IoAnQ ೧ 8pongiLla )_ ( للط لللمم اعرم mukceiuLor organuims org syslom_৮ prexnk wbld ConcQ Ilcten tronspont 01 hop cion /and excretion o @huton r೮೦ i೧ Digestioo fs lotroce tlulo ع spongio,Gibres_or-Spiculea & Skelekool mmode ob not sepecate ) Cexe Hemaphrodiel QL FeriuboD &sA(oterocQ develanh preLent) ( LccUcL_Stcge is Ind(roue Syoon , Spongillc bcH Spoge ೭೦ PHVUN CDELENTEBBTA CCNIDBRIB moskut swlDomio Iheg ore {oe sessyel moul ue ) 0 Mulieuloک okaorisms levell w &AuQ 09` Aada[U The ore diploblooticzand Slmmak?t onidoudtoz Cnioob@b pue೨enh ٥ oD kotack Cod9 and |e Cnfoblasbs /e_uoed_kur_ccboraq-e_sdebente] 0 Q0d Ihe ophura n pr 8 Dioetioo intrateuulcc & extq ckiulaC , and howe LaLm CrboDote CeaAl SRelet Cniclariq xbubit மooy bmப ೧ bolyp abd meduAQ _ Polyps & %LQ ZrdcQ bom_LRe ODd hudro ) adamstql PHYUM PRIEERD They ஈீD $ cewh catel Qute goneollப IoAnQ ೧ 8pongiLla )_ ( للط لللمم اعرم mukceiuLor organuims org syslom_৮ prexnk wbld ConcQ Ilcten tronspont 01 hop cion /and excretion o @huton r೮೦ i೧ Digestioo fs lotroce tlulo ع spongio,Gibres_or-Spiculea & Skelekool mmode ob not sepecate ) Cexe Hemaphrodiel QL FeriuboD &sA(oterocQ develanh preLent) ( LccUcL_Stcge is Ind(roue Syoon , Spongillc bcH Spoge ೭೦ PHVUN CDELENTEBBTA CCNIDBRIB moskut swlDomio Iheg ore {oe sessyel moul ue ) 0 Mulieuloک okaorisms levell w &AuQ 09` Aada[U The ore diploblooticzand Slmmak?t onidoudtoz Cnioob@b pue೨enh ٥ oD kotack Cod9 and |e Cnfoblasbs /e_uoed_kur_ccboraq-e_sdebente] 0 Q0d Ihe ophura n pr 8 Dioetioo intrateuulcc & extq ckiulaC , and howe LaLm CrboDote CeaAl SRelet Cniclariq xbubit மooy bmப ೧ bolyp abd meduAQ _ Polyps & %LQ ZrdcQ bom_LRe ODd hudro ) adamstql - ShareChat")
