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ISC Class XI Notes 2024 : Mathematics (Vidyatree Modern World College - The Modern School, Lucknow)

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1. Linear Function Example: f(x) == mx + c. Graph: A strai ght line with slope m and y-inte rcep t c. Domain and Range: Both are all real numbers (JR). 2. Quadratic Function f (x) 2 == ax +bx + c. Example: Graph: A parab ola, which open s upwa rds if a Dom ain: All real numb ers. > 0 and down ward s if a < 0. rds, the range is [k, oo ); Range: Depe nds on the vertex. If the parab ola open s upwa inate of the if it open s down ward s, the range is ( -oo, k], where k is they- coord verte x. 3. Expo nent ial Function f ( x) == ax (with a > 1). Exam ple: aches zero for negative Grap h: A curve that increases rapid ly for positive x and appro x. Dom ain: All real numbers. Range: (0, oo ). 4. Loga rithm ic Func tion J(x) == loga( x). Example: aches negat ive infinit y Graph: A curve that increa ses slowl y for positi ve x and appro as x appro aches zero from the right. (0, CX) ). Doma in: Rang e: All real numb ers. 5. Sign um Function Graph: A horizontal line at the origi n for x == y == l for x > 0, at y == -l for x < 0, and a poin t at 0. Dom ain: All real numbers. Rang e: {-1 , 0, 1}. 6. Absolute Value Function f (x) \xi. Example: Grap h: AV-s hape d graph that is linear with a slope of 1 for x 0. Dom ain: All real numbers. Range: == > 0 and -1 for x (0, oo ). 7. Piece wise Func tion Definition: A funct ion defin ed by differ ent expre ssion s for differ ent interv als of the doma in. Exam ple: f (x) == { x2 if X -x if X > 0, < 0. Graph: The graph will comb ine the graph s of the indivi dual functi ons, each restri cted to their respe ctive intervals. Doma in and Range: Depe nds on the indivi dual pieces. < Un de rst an din g Functions in Mathematics 1. Basic Definitions Fun ction : A func tion is a rela tion betw een two sets, whe re each elem ent of the first set (called the dom ain) is related to exactly one elem ent of the second set (called the codomain). If is a func tion from X to Y, we write X f f : Y. Domain: The dom ain of a func tion is the set of all possible inpu ts for which the func tion is defin ed. For example, if f ( x) == tQe dom ain is [O, oo) because the square root func tion is only defin ed for non- nega tive numbers. Jx, Codomain: The codo main is the set of all possible outp uts that coul d pote ntial ly be prod uced by the func tion. It includes all values that the func tion migh t map to, but not necessarily all values that are actually achieved. Range: The range of a function is the set of all actu al outp uts (i.e., the values that f (x) actually takes when x varies over the domain). For example, for f (x) == 2 x , the range is [O, oo) when the domain is all real num bers. Image: The image of an element x in the domain under the function f is the corresponding element yin the codomain such that y == f (x ). Pre-image: The pre-image of an element y in the codo main is the set of all elements in the domain that map toy. If f (x) == y, then xis the pre-image of y. 2. Types of Functions A. Injective (One -to-O ne) Function Defin ition : A funct ion f : X Y is injec tive if for every /(x2 ) impli es X1 == X2, x 1 , x 2 E X, f ( x 1 ) == Characteristics: Diffe rent elem ents of the doma in map to differ ent elem ents of the codo main . The funct ion has a left inverse (i.e., an inverse junct ion that maps back to the doma in). Grap hicall y, an inject ive funct ion passes the horizontal line test- no horiz ontal line inters ects the graph at more than one point . Example: same f (x) == 2x + 3 is injective because no two different x values produce the f(x). B. Surje ctive (Onto ) Function least one x E f : X Y is surjective if for every y X such that f (x) == y. Definition: A funct ion E Y, there exists at Characteristics: The range of the funct ion is equal to its codomain. The funct ion has a right inverse. Graphically, every elem ent of the codomain has at least one corre sponding point on the graph. f (x) == x 3 is surjective when both the domain and codomain are IR, since every real numb er can be writte n as a cube of some real numb er. Example: C. Bijective Function De fin itio n: A fun ctio n f : X Y is bije ctiv e if it is bo th inje ctiv e and surjective. This means each ele me nt of the do ma in maps to a uni que ele me nt in the cod om aii and eve ry ele me nt of the cod om ain has a pre -im age . Characteristics: Bijective fun ctio ns are bo th on e-t o-o ne and ont o. They have bo th left and rig ht invers es, me ani ng the fun ctio n is invertible . Graphically, a bijective fun ctio n pas ses bo th tkle hor izo nta l line test and the vertical line test. Example: f ( x) == 2x 1- 1 (y) == y 2 3. + 3 is bijective when defined from JR to JR, and its inverse D. Co nst ant Function Definition: A fun ctio n tha t maps eve ry element of the dom ain to the sam e single ele me nt of the codomain. Characteristics: The graph of a constant function is a horizontal line. The range consists of a single value. Constant functions are not injective (unless the domain is a single point), but the y can be surjective depending on the codomain. Example: f ( x) === 5 for all x E IR. E. Ide nti ty Fun ctio n Definition: A function tha t maps every element to itself. Formally, XE X. f (x) = x for all Characteristics: The gra ph of the ide ntit y fun ctio n is a stra igh t line thro ugh the orig in wit h a slo pe of 1. It is bije ctiv e, me ani ng it is bot h inje ctiv e and sur ject ive. Example: f ( x) = x on the dom ain -.!, is F. Polynomial Function Defini tion: A functio n that can be expressed as + a 1 x + a 0, where an, an f (x) == an xn + an-l xn-l + 1 , ... , a 0 are constants. Characteristics: The degree of the polynomial determines the general shape of the graph. Polynomials of even degree have graphs that tend to infinity in both directions, while those of odd degree tend to infinity in one direction and negative infinity in the other. Domain: All real numbers. Example: f (x) == x 2 - 4x + 7. G. Rational Function Definit ion: A functio n that is the ratio of two polyno mials, and q( x) are polyno mials and q( x) -=/- f (x) == :/;j, where p(x) 0. Characteristics: The graph can have vertica l asymp totes where the denom inator equals zero. Horizo ntal asymp totes depen d on the degree s of the numer ator and denom inator. Domai n: All real numbe rs except where the denom inator is zero. Example: f ( x) == x~ 2 . H. Expon ential Function Definition: A functio n of the form Characteristics: f (x) == ax, where a is a positiv e constant. The graph rises rapidly for positive x and approaches zero for negative x. Domain: All real numbers. Range:(O,CX)). Examp le: J(J;) - 2 ,.. I. Logarithmic Function Definition: The inverse of an exponential function, y == ax. The logar ithmi c function is x == loga (y ). Characteristics: The graph increases slowly for positive x and approaches negative infinity as x approaches zero from the right. (0, oo ). Domain: Range: All real numbers. Example: f (x) == log( x ). J. Trigo nome tric Functions Examples: Sine Function f ( x) == sin( x ): Graph: A perio dic wave oscill ating betwe en -1 and 1. Domain: All real numb ers. Range: [-1, Cosin e Funct ion l]. f (x) == cos( x ): Graph: Simila r to the sine functi on, but starts at 1 when x Doma in: All real numb ers. Range: [-1, l]. Tange nt Function == 0. f (x) == tan(x ): Graph: Has vertic al asymp totes where cos( x) == 0, with the functi on appro achin g infinit y betwe en these points . Doma in: All real numb ers excep t x Range : All real numb ers. = ; + nn, where n is an intege r. K. Sig num Fun ctio n Definition: A func tion that extracts the sign of a real number. Characteristics: Graph: A hori zon tal line at at the orig in for x == Domain: All real numbers. Range: Example: > 0, at y == -1 for x < 0, and a poin t 0. {-1 ,0,1 }. -1 y == 1 for x sgn ( x) == 0 1 if X < 0, if X == 0, if X > 0. L. Absolute Value Function Definition: A func tion that returns the non -neg ativ e value of a real num ber, rega rdle ss of its sign. Characteristics: Graph: AV- sha ped grap h with a vertex at the orig in. Domain: All real numbers. Range: Example: [O, oo). f(x ) == !xi. 3. Graphical Representation of Functions The graph of a function is a visual representation of the set of ordered pairs ( x, f (x)) the Cartesian plane. The graph provides a powerful way to understand the behavior of functions, including their domains, ranges, and key characteristics such as symmetry, intercepts, and asymptotes. A. Linear Function f (x) == mx + c. Example: Graph: A straight line with slope m and y-intercepJ c. Domain and Range: Both are all real numbers (]R). B. Quadratic Function Example: f(x) == ax 2 +bx+ c. Graph: A parabola, which opens upwards if a Domain: All real numbers. Range: Depends on the vertex. If the parabola opens upwards, the range is if 1t opens downwards, the range is vertex. > 0 and downwards if a < O. (-oo , k] , where k , s th [k, oo ); d. e y-coor ,nate of the C. Exponential Function f (x) == ax (with a > 1). Example: d Graph: A curve that increases rapidly for positive x an approaches zero for negative x. Domain: All real numbers. Range: (0, oo ). D. Logarithmic Function Exam p Ie: Graph A f ( x) == log,, ( x). - . . curve that increases slowly for positive a and a pproaches negative infinity . I as x approaches zero from ti le (0, 00 ). Domain: Range: All real numbers. rig lt. in E. Signum Function Graph: A horizontal line at y == 1 for x the origin for x == 0. Domain: All real numbers. Range: {-1, 0, 1}. > 0, at y == -1 for x < 0, and a poin t at F. Abso lute Value Function Ix I. Example: f ( x) == Graph: AV-s hape d graph that is linear with a slope of 1 for x 0. Dom ain: All real numbers. Range: > 0 and -1 for x < [O, oo ). G. Piecewise Function Graph: The graph will comb ine the graph s of the indivi dual funct ions, each restri cted to their respe ctive interv als. Dom ain and Rang e: Depe nds on the indivi dual pieces. 4. lmp o rtan t Prop ertie s and Theo rem s Rela ted to Func tions 1 A. Com posit ion of Functions Defin ition: Given two funct ions f : X Y and g : Y Z, their comp ositio n go f: X Z is defin ed by (go J)(x ) == g(f(x )). Characteristics: The doma in of g o f is the set of all x in X such that f ( x) is in the doma in of g. Comp ositio n is associative but not necessarily comm utativ e. B. Inverse Functions Definition: If f : X Y is a bijection, its inverse f f l (y) == X if f ( X) == y. 1 : Y X is defined by Characteristics: The graph of f(f- 1 (y)) f- 1 is the reflection of the graph off across the line y == x. == y and f 1 (/(x) ) == X. " C. Even and Odd Functions Even Functi on: Defini tion: A functio n is even if f(x) == f(-x) for all x in the domai n. Graph: Symm etric about the y-axis. Example: f (x) == x 2 . Odd Function: Definition: A functio n is odd if f Graph: Symm etric about the origin. Example: f ( x) == x 3 (-x) == -f (x) for all x in the domain .

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