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1 SHORT REVISION (FUNCTIONS) THINGS TO REMEMBER :. GENERAL DEFINITION : If to every value (Considered as real unless otherwise stated) of a variable which belongs to some collection (Set) E there corresponds one and only one finite value of the quantity y then y is said to be a function (Single valued) of or a dependent variable defined on the set E ; is the argument or independent variable. If to every value of belonging to some set E there corresponds one or several values of the variable y then y is called a multiple valued function of defined on E.Conventionally the word "FUNCTION is used only as the meaning of a single valued function if not otherwise stated. input Pictorially : f ( ) y output y is called the image of & is the pre-image of y under f. Every function from A B satisfies the following conditions. (i) f A B (ii) a A (a f(a)) f and (iii) (a b) f & (a c) f b = c. DOMAIN CODOMAIN & RANGE OF A FUNCTION : Let f : A B then the set A is known as the domain of f & the set B is known as co-domain of f. The set of all f images of elements of A is known as the range of f. Thus : Domain of f = {a a A (a f(a)) f} Range of f = {f(a) a A f(a) B} It should be noted that range is a subset of codomain. If only the rule of function is given then the domain of the function is the set of those real numbers where function is defined. For a continuous function the interval from minimum to maimum value of a function gives the range. 3. IMPORTANT TYPES OF FUNCTIONS : (i) POLYNOMIAL FUNCTION : If a function f is defined by f () = a 0 n + a n + a n a n + a n where n is a non negative integer and a 0 a a... a n are real numbers and a 0 0 then f is called a polynomial function of degree n. NOTE : (a) A polynomial of degree one with no constant term is called an odd linear function. i.e. f() = a a 0 (b) There are two polynomial functions satisfying the relation ; f().f(/) = f() + f(/). They are : (i) f() = n + & (ii) f() = n where n is a positive integer. (ii) ALGEBRAIC FUNCTION : y is an algebraic function of if it is a function that satisfies an algebraic equation of the form P 0 () y n + P () y n P n () y + P n () = 0 Where n is a positive integer and P 0 () P ()... are Polynomials in. e.g. y = is an algebraic function since it satisfies the equation y² ² = 0. Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called TRANSCEDENTAL FUNCTION. (iii) FRACTIONAL RATIONAL FUNCTION : A rational function is a function of the form. y = f () = g ( ) where h( ) g () & h () are polynomials & h () 0. (IV) EXPONENTIAL FUNCTION : A function f() = a = e ln a (a > 0 a R) is called an eponential function. The inverse of the eponential function is called the logarithmic function. i.e. g() = log a. Note that f() & g() are inverse of each other & their graphs are as shown. Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 4 of 4

2 (v) ABSOLUTE VALUE FUNCTION : A function y = f () = is called the absolute value function or Modulus function. It is defined as if 0 : y = if 0 (vi) SIGNUM FUNCTION : A function y= f () = Sgn () is defined as follows : for 0 y = f () = 0 for 0 for 0 It is also written as Sgn = / ; 0 ; f (0) = 0 (vii) GREATEST INTEGER OR STEP UP FUNCTION : The function y = f () = [] is called the greatest integer function where [] denotes the greatest integer less than or equal to. Note that for : < 0 ; [] = 0 < ; [] = 0 < ; [] = < 3 ; [] = and so on. Properties of greatest integer function : y (a) [] < [] + and graph of y = [] 3 < [] 0 [] < (b) [ + m] = [] + m if m is an integer. º (c) [] + [y] [ + y] [] + [y] + º (d) y = (0 ) f() = a a > )45º ( 0) + g() = log a [] + [ ] = 0 if is an integer = otherwise. (viii) FRACTIONAL PART FUNCTION : It is defined as : g () = {} = []. e.g. the fractional part of the no.. is. = 0. and the fractional part of 3.7 is 0.3. The period of this function is and graph of this function is as shown. 4. DOMAINS AND RANGES OF COMMON FUNCTION : Function Domain Range (y = f () ) (i.e. values taken by ) (i.e. values taken by f () ) A. Algebraic Functions y = if < 0 (i) n (n N) R = (set of real numbers) R if n is odd R + {0} if n is even + y = (0 ) f() = a 0 < a < )45º ( 0) g() = log a y O y = if > 0 º 3 º 3 º 3 y > y = Sgn graph of y = {} º º º º Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 5 of 4 (ii) n (n N) R {0} R {0} if n is odd

3 R + if n is even FREE Download Study Package from website: & (iii) (iv) / n (n N) R if n is odd R if n is odd R + {0} if n is even R + {0} if n is even / B. Trigonometric Functions n (n N) R {0} if n is odd R {0} if n is odd R + if n is even R + if n is even (i) sin R [ + ] (ii) cos R [ + ] (iii) tan R (k + ) ki (iv) sec R (k + ) ki ( ] [ ) (v) cosec R k k I ( ] [ ) (vi) cot R k k I R C. Inverse Circular Functions (Refer after Inverse is taught ) (i) sin [ + ] (ii) cos [ + ] [ 0 ] (iii) tan R (iv) cosec ( ] [ ) { 0 } (v) sec ( ] [ ) [ 0 ] (vi) cot R ( 0 ) R Function Domain Range (y = f () ) (i.e. values taken by ) (i.e. values taken by f () ) D. Eponential Functions (i) e R R + (ii) e / R { 0 } R + { } (iii) a a > 0 R R + (iv) a / a > 0 R { 0 } R + { } E. Logarithmic Functions (i) log a (a > 0 ) (a ) R + R (ii) log a = log a (a > 0 ) (a ) R + { } R { 0 } Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 6 of 4 F. Integral Part Functions Functions

4 (i) [ ] R I (ii) [] R [0 ) n n I {0} FREE Download Study Package from website: & G. Fractional Part Functions (i) { } R [0 ) (ii) {} H. Modulus Functions R I ( ) (i) R R + { 0 } (ii) I. Signum Function J. Constant Function R { 0 } R + sgn () = 0 R { 0 } = 0 = 0 say f () = c R { c } 5. EQUAL OR IDENTICAL FUNCTION : Two functions f & g are said to be equal if : (i) The domain of f = the domain of g. (ii) The range of f = the range of g and (iii) f() = g() for every belonging to their common domain. eg. f() = & g() = are identical functions. 6. CLASSIFICATION OF FUNCTIONS : One One Function (Injective mapping) : A function f : A B is said to be a oneone function or injective mapping if different elements of A have different f images in B. Thus for A & f( ) f( ) B f( ) = f( ) = or f( ) f( ). Diagramatically an injective mapping can be shown as Note : (i) Any function which is entirely increasing or decreasing in whole domain then f() is oneone. (ii) If any line parallel to ais cuts the graph of the function atmost at one point then the function is oneone. Many one function : A function f : A B is said to be a many one function if two or more elements of A have the same f image in B. Thus f : A B is many one if for ; A f( ) = f( ) but. OR Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 7 of 4

5 Diagramatically a many one mapping can be shown as FREE Download Study Package from website: & Note : (i) Any continuous function which has atleast one local maimum or local minimumthen f() is manyone. In other words if a line parallel to ais cuts the graph of the function atleast at two points then f is manyone. (ii) If a function is oneone it cannot be manyone and vice versa. OR Onto function (Surjective mapping) : If the function f : A B is such that each element in B (codomain) is the f image of atleast one element in A then we say that f is a function of A 'onto' B. Thus f : A B is surjective iff b B some a A such that f (a) = b. Diagramatically surjective mapping can be shown as OR Note that : if range = codomain then f() is onto. Into function : If f : A B is such that there eists atleast one element in codomain which is not the image of any element in domain then f() is into. Diagramatically into function can be shown as OR Note that : If a function is onto it cannot be into and vice versa. A polynomial of degree even will always be into. Thus a function can be one of these four types : (a) (b) (c) (d) Note : (i) (ii) oneone onto (injective & surjective) oneone into (injective but not surjective) manyone onto (surjective but not injective) manyone into (neither surjective nor injective) If f is both injective & surjective then it is called a Bijective mapping. The bijective functions are also named as invertible non singular or biuniform functions. If a set A contains n distinct elements then the number of different functions defined from A A is n n & out of it n! are one one. Identity function : The function f : A A defined by f() = A is called the identity of A and is denoted by I A. It is easy to observe that identity function is a bijection. Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 8 of 4

6 Constant function : A function f : A B is said to be a constant function if every element of A has the same f image in B. Thus f : A B ; f() = c A c B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one onto or into. 7. ALGEBRAIC OPERATIONS ON FUNCTIONS : If f & g are real valued functions of with domain set A B respectively then both f & g are defined in A B. Now we define f + g f g (f. g) & (f/g) as follows : (i) (f ± g) () = f() ± g() (ii) (f g) () = f() g() (iii) f g () = f ( ) g( ) domain is { A B s. t g() 0}. 8. COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS : Let f : A B & g : B C be two functions. Then the function gof : A C defined by (gof) () = g (f()) A is called the composite of the two functions f & g. Diagramatically f ( ) g (f()). Thus the image of every A under the function gof is the gimage of the fimage of. Note that gof is defined only if A f() is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f & g the range of f must be a subset of the domain of g. PROPERTIES OF COMPOSITE FUNCTIONS : (i) The composite of functions is not commutative i.e. gof fog. (ii) The composite of functions is associative i.e. if f g h are three functions such that fo (goh) & (fog) oh are defined then fo (goh) = (fog) oh. (iii) The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined then gof is also a bijection. 9. HOMOGENEOUS FUNCTIONS : A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables. For eample y y is homogeneous in & y. Symbolically if f (t ty) = tn. f ( y) then f ( y) is homogeneous function of degree n. 0. BOUNDED FUNCTION : A function is said to be bounded if f() M where M is a finite quantity.. IMPLICIT & EXPLICIT FUNCTION : A function defined by an equation not solved for the dependent variable is called an IMPLICIT FUNCTION. For eg. the equation 3 + y 3 = defines y as an implicit function. If y has been epressed in terms of alone then it is called an EXPLICIT FUNCTION.. INVERSE OF A FUNCTION : Let f : A B be a oneone & onto function then their eists a unique function g : B A such that f() = y g(y) = A & y B. Then g is said to be inverse of f. Thus g = f : B A = {(f() ) ( f()) f}. PROPERTIES OF INVERSE FUNCTION : (i) The inverse of a bijection is unique. (ii) If f : A B is a bijection & g : B A is the inverse of f then fog = I B and gof = I A where I A & I B are identity functions on the sets A & B respectively. Note that the graphs of f & g are the mirror images of each other in the line y =. As shown in the figure given below a point ( 'y ' ) corresponding to y = ( >0) changes to (y ' ' ) corresponding to y the changed form of = y. Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 9 of 4

7 (iii) The inverse of a bijection is also a bijection. (iv) If f & g are two bijections f : A B g : B C then the inverse of gof eists and (gof) = f o g. 3. ODD & EVEN FUNCTIONS : If f () = f () for all in the domain of f then f is said to be an even function. e.g. f () = cos ; g () = ² + 3. If f () = f () for all in the domain of f then f is said to be an odd function. e.g. f () = sin ; g () = 3 +. NOTE : (a) f () f () = 0 => f () is even & f () + f () = 0 => f () is odd. (b) A function may neither be odd nor even. (c) Inverse of an even function is not defined. (d) Every even function is symmetric about the yais & every odd function is symmetric about the origin. (e) Every function can be epressed as the sum of an even & an odd function. f( ) f( ) f( ) f( ) e.g. f( ) (f) (g) The only function which is defined on the entire number line & is even and odd at the same time is f() = 0. If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd. 4. PERIODIC FUNCTION : A function f() is called periodic if there eists a positive number T (T > 0) called the period of the function such that f ( + T) = f() for all values of within the domain of. e.g. The function sin & cos both are periodic over & tan is periodic over. NOTE : (a) f (T) = f (0) = f (T) where T is the period. (b) Inverse of a periodic function does not eist. (c) Every constant function is always periodic with no fundamental period. (d) If f () has a period T & g () also has a period T then it does not mean that f () + g () must have a period T. e.g. f () = sin+ cos. (e) If f() has a period p then f ( ) and f ( ) also has a period p. (f) if f() has a period T then f(a + b) has a period T/a (a > 0). 5. GENERAL : If y are independent variables then : (i) f(y) = f() + f(y) f() = k ln or f() = 0. (ii) f(y) = f(). f(y) f() = n n R (iii) f( + y) = f(). f(y) f() = a k. (iv) f( + y) = f() + f(y) f() = k where k is a constant. EXERCISE Q. Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f () = cos 6 (ii) f () = log 7 log 5 log 3 log ( ) (iii) f () = l n 5 4 (iv) f () = Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 0 of 4 (v) y = log0 sin( 3) 6 (vi) f () = log0 log00

8 (vii) f () = ln ( ) (viii) f () = log 4 (i) f( ) 9 (i) f() = log (cos ) (ii) f () = (iii) f() = log log [] 5 (v) f() = log sin /3 4 log 00 (vi) f() = log / sin (vii) f () = [ ] + log {} ( 3 + 0) + (viii) f () = (i) If f() = () f () = ( 3 0). ln ( 3) cos (iv) f() = ( { } 5) 3 0 log ( ) [ ] + log log log 4 log log sec(sin ) 7 (5 6 ) l n + (7 5 ) + ln 5 4 & g() = + 3 then find the domain of f g (). Q. Find the domain & range of the following functions. ( Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) y = log (sin cos ) 3 5 (iv) f () = (ii) y = (v) y = (iii) f() = (vi) f () = log (cosec - ) ( [sin ] [sin ] ) (vii) f () = 5 Q.3 Draw graphs of the following function where [] denotes the greatest integer function. (i) f() = + [] (ii) y = () [] where = [] + () & > 0 & 3 (iii) y = sgn [] (iv) sgn ( ) Q.4 Classify the following functions f() definzed in R R as injective surjective both or none. (a) f() = (b) f() = (c) f() = ( + + 5) ( + 3) Q.5 Let f() =. Let f () denote f [f ()] and f () denote f [f {f()}]. Find f () where n is a natural 3 3n number. Also state the domain of this composite function. 5 Q.6 If f() = sin² + sin² cos cos and g then find (gof)() Q.7 The function f() is defined on the interval [0]. Find the domain of definition of the functions. (a) f (sin ) (b) f (+3) Q.8(i) Find whether the following functions are even or odd or none (a) f() = log a (b) f() = a (c) f() = sin + cos (d) f() = sin 3 (e) f()= sin cos (f) f() = Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page of 4

9 (g) f()= e (h) f() = [(+)²] /3 + [( )²] /3 (ii) If f is an even function defined on the interval (5 5) then find the 4 real values of satisfying the equation f () = f.. Q.9 Write eplicitly functions of y defined by the following equations and also find the domains of definition of the given implicit functions : (a) y = 0 (b) + y= y Q.0 Show if f() = n a n > 0 n n N then (fof) () =. Find also the inverse of f(). Q. (a) Represent the function f() = 3 as the sum of an even & an odd function. (b) For what values of p z the function f() = n p n N is even. Q. A function f defined for all real numbers is defined as follows for 0 : f ( ) [ How is f defined for 0 if : (a) f is even (b) f is odd? Q.3 If f () = ma 0 for > 0 where ma (a b) denotes the greater of the two real numbers a and b. Define the function g() = f(). f and plot its graph. Q.4 T h e f u n c t i o n f () has the property that for each real number in its domain / is also in its domain and f () + f =. Find the largest set of real numbers that can be in the domain of f ()? Q.5 Compute the inverse of the functions: (a) f() = ln (b) f() = (c) y = Q.6 A function f : 3 4 defined as f() = +. Then solve the equation f () = f (). Q.7 Function f & g are defined by f() = sin R ; g() = tan R K where K I. Find (i) periods of fog & gof. (ii) range of the function fog & gof. Q.8 Find the period for each of the following functions : (a) f()= sin 4 + cos 4 (b) f() = cos (c) f()= sin+cos (d) f()= cos 3 5 sin 7. Q.9 Prove that the functions ; (a) f() = cos (b) f() = sin (c) f() = + sin (d) f() = cos are not periodic. Q.0 Find out for what integral values of n the number 3 is a period of the function : f() = cos n. sin (5/n). EXERCISE Q. Let f be a oneone function with domain {yz} and range {3}. It is given that eactly one of the following statements is true and the remaining two are false. f() = ; f(y) ; f(z). Determine f () Q. Solve the following problems from (a) to (e) on functional equation. (a) The function f () defined on the real numbers has the property that f ( ) f () f = f () for all in the domain of f. If the number 3 is in the domain and range of f compute the value of f (3). Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page of 4

10 (b) Suppose f is a real function satisfying f ( + f ()) = 4 f () and f () = 4. Find the value of f (). FREE Download Study Package from website: & (c) L e t 'f' be a function defined from R + R +. If [ f (y)] = (d) f () = 6 find the value of f (50). Let f () be a function with two properties (i) for any two real number and y f ( + y) = + f (y) and (ii) f (0) =. Find the value of f (00). f (y) for all positive numbers and y and (e) Let f be a function such that f (3) = and f (3) = + f (3 3) for all. Then find the value of f (300). Q.3(a) A function f is defined for all positive integers and satisfies f() = 005 and f()+ f() f(n) = n f(n) for all n >. Find the value of f(004). (b) If a b are positive real numbers such that a b = then find the smallest value of the constant L for which a b < L for all > 0. (c) Let f () = + k ; k is a real number. The set of values of k for which the equation f () = 0 and f f () = 0 have same real solution set. (d) If f ( + ) = then find the sum of the roots of the equation f () = 0. a b 5 Q.4 Let f () = for real a b and c with a 0. If the vertical asymptote of y = f () is = and the 4 c 4 vertical asymptote of y = f () is = 4 3 find the value(s) that b can take on. Q.5 A function f : R R satisfies the condition f () + f ( ) = 4. Find f () and its domain and range. Q.6 Suppose p() is a polynomial with integer coefficients. The remainder when p() is divided by is and the remainder when p() is divided by 4 is 0. If r () is the remainder when p() is divided by ( )( 4) find the value of r (006). e Q.7 Prove that the function defined as f () = {} ln{} ln{} {} where ever f () is odd as well as even. ( where {} denotes the fractional part function ) otherwise then Q.8 In a function f() + f f sin 4 = 4 cos + cos Prove that (i) f() + f(/) = and (ii) f() + f() = 0 Q.9 A function f defined for all y R is such that f () = ; f () = 8 & f ( + y) k y = f () + y where k is some constant. Find f () & show that : f ( + y) f = k for + y 0. y eists Q.0 Let f be a real valued function defined for all real numbers such that for some positive constant a the equation f( a) f( ) f( ) holds for all. Prove that the function f is periodic. Q. If f () = g () = 3 Then find fog () & gof (). Draw rough sketch of the graphs of fog () & gof (). Q. Find the domain of definition of the implicit function defined by the implicit equation 3 y + 4 = 4. it Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 3 of 4

11 Q.3 Let {} & [] denote the fractional and integral part of a real number respectively. Solve 4{}= + [] FREE Download Study Package from website: & 9 Q.4 Let f () = 9 3 then find the value of the sum f f + f f Q.5 Let f () = ( + )( + )( + 3)( + 4) + 5 where [ 6 6]. If the range of the function is [a b] where a b N then find the value of (a + b). Q.6 Find a formula for a function g () satisfying the following conditions (a) domain of g is ( ) (b) range of g is [ 8] (c) g has a period and (d) g () = Q.7 The set of real values of '' satisfying the equality + = 5 (where [ ] denotes the greatest integer b b function) belongs to the interval a where a b c N and is in its lowest form. Find the value of c c a + b + c + abc. Q.8 Find the set of real for which the function f() = denotes the greatest integer function. is not defined where [] Q.9 A is a point on the circumference of a circle. Chords AB and AC divide the area of the circle into three equal parts. If the angle BAC is the root of the equation f () = 0 then find f (). Q.0 If for all real values of u & v f(u) cos v = f(u + v) + f(u v) prove that for all real values of (i) f() + f( ) = a cos (ii) f( ) + f( ) = 0 (iii) f( ) + f() = b sin. Deduce that f() = a cos b sin a b are arbitrary constants. EXERCISE 3 Q. If the functions f g h are defined from the set of real numbers R to R such that ; f ()= g () = h () = 0 if if 0 ; then find the composite function ho(fog) & determine 0 whether the function (fog) is invertible & the function h is the identity function. [REE '97 6] Q.(a) If g (f()) = sin & f (g()) = sin then : (A) f() = sin g() = (C) f() = g() = sin (b) If f() = 3 5 then f () (A) is given by 3 5 (B) f() = sin g() = (D) f & g cannot be determined (B) is given by 5 3 (C) does not eist because f is not oneone (D) does not eist because f is not onto [JEE'98 + ] Q.3 If the functions f & g are defined from the set of real numbers R to R such that f() = e g() = 3 then find functions fog & gof. Also find the domains of functions (fog) & (gof). [ REE '98 6 ] Q.4 If the function f : [ ) [ ) is defined by f() = ( ) then f () is : [ JEE '99 ] (A) ( ) (B) log (C) 4 4 log (D) not defined Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 4 of 4

12 Q.5 The domain of definition of the function y () given by the equation + y = is : (A) 0 < (B) 0 (C) < 0 (D) < < Q.6 Given = { 3 4} find all oneone onto mappings f : X X such that f () = f () and f (4) 4. [ REE out of 00 ] 0 Q.7(a) Let g () = + [ ] & f () = 0 0. Then for all f (g ()) is equal to 0 (A) (B) (C) f () (D) g () (b) If f : [ ) [ ) is given by f () = + then f () equals (A) 4 (B) (C) 4 (D) 4 log ( 3) (c) The domain of definition of f () = 3 is : (d) (A) R \ { } (B) ( ) (C) R\{ 3} (D) ( 3 ) \ { } Let E = { 3 4 } & F = { }. Then the number of onto functions from E to F is (A) 4 (B) 6 (C) (D) 8 (e) Let f () =. Then for what value of is f (f ()) =? (A) (B) (C) (D). Q.8(a) Suppose f() = ( + ) for >. If g() is the function whose graph is the reflection of the graph of f() with respect to the line y = then g() equals (A) > 0 (B) > (C) > (D) ( ) > 0 (b) Let function f : R R be defined by f () = + sin for R. Then f is (A) one to one and onto (B) one to one but NOT onto (C) onto but NOT one to one (D) neither one to one nor onto Q.9(a) Range of the function f () = is (A) [ ] (B) [ ) 7 (C) 7 (D) 3 3 (b) Let f () = defined from (0 ) [ 0 ) then by f () is (A) one- one but not onto (B) one- one and onto (C) Many one but not onto (D) Many one and onto [JEE 003 (Scr)3+3] Q.0 Let f () = sin + cos g () =. Thus g ( f () ) is invertible for (A) 0 (B) (C) (D) [JEE 004 (Screening)] Q.(a) If the functions f () and g () are defined on R R such that f () = 0 rational g () = 0 irrational irrational rational then (f g)() is (A) one-one and onto (B) neither one-one nor onto (C) one-one but not onto (D) onto but not one-one (b) X and Y are two sets and f : X Y. If {f (c) = y; c X y Y} and {f (d) = ; d Y X} then the true statement is (A) f f (b) b (B) f f (a) a (C) f f (b) b b y (D) f f (a) a a [JEE 005 (Scr.)] Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 5 of 4

13 ANSWER KEY FUNCTIONS EXERCISE Q. (i) (ii) 4 ( )(iii) ( 3] (iv) ( ) [0 ) (v) (3 < < 3 ) U (3 < 4) (vi) (vii) ( < < /) U ( > ) (viii) (i) (3 ] U {0} U [ 3 ) 5 () { 4 } [ 5 ) (i) (0 /4) U (3/4 ) U { : N } (ii) (iii) [ 3 ) [ 34) (iv) (v) K < < (K + ) but where K is nonnegative integer 5 (vi) { 000 < 0000} (vii) ( ) U ( 0) U ( ) (viii) ( ) (i) ( 3) (3 ] [4 ) Q. (i) D : R R : [0 ] (ii) D = R ; range [ ] (iii) D : { R ; 3 ; } R : {f()f()r f() /5 ; f() } (iv) D : R ; R : ( ) (v) D : R : 3 6 (vi) D : (n (n + )) n n n n I and R : log a ; a (0 ) {} Range is ( ) {0} (vii) D : [ 4 ) {5}; R : Q.4 (a) neither surjective nor injective (b) surjective but not injective (c) neither injective nor surjective Q.5 f 3n () = ; Domain = R {0 } Q.6 Q.7 (a) K K + where K I (b) [3/ ] Q.8 (i) (a) odd (b) even (c) neither odd nor even (d) odd (e) neither odd nor even (f) even (g) even (h) even; (ii) Q.9 (a) y = log (0 0 ) < < (b) y = /3 when < < 0 & y = when 0 < + Q.0 f () = (a n ) /n Q. (a) f() = for < & for 0; (b) f() = for < and for 0 Q.3 g() if 0 if Q.4 { } Q.5 (a) e e log ; (b) log ; (c) log Q.6 = Q.7 (i) period of fog is period of gof is ; (ii) range of fog is [ ] range of gof is [tan tan] Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 6 of 4

14 Q.8 (a) / (b) (c) / (d) 70 Q.0 ± ± 3 ± 5 ± 5 FREE Download Study Package from website: & Q. f () = y Q. (a) 3/4 (b) 64 (c) 30 (d) 0 (e) 5050 Q.3 (a) (b) (c) [0 4) (d) 5 00 EXERCISE 5 Q 4. b can be any real number ecept Q5. f () = D = R ; range =( ] 4 Q Q 9. f () = Q. fog () = ( ) Q. 0 0 fof () = ; gof () = ; gog () = Q.3 = 0 or 5/3 Q Q Q.6 g () = sin(n + 4) n I Q.7 0 Q 8. (0 ) {... } ( 3) Q 9. f () = sin + 3 EXERCISE 3 Q. (hofog)() = h( ) = for R Hence h is not an identity function fog is not invertible Q. (a) A (b) B Q.3 (fog) () = e 3 ; (gof) () = 3 e ; Domain of (fog) = range of fog = (0 ); Domain of (gof) = range of gof = ( ) Q.4 B Q.5 D Q.6 {( ) ( 3) (3 4) (4 )} ; {( ) ( 4) (3 ) (4 3)} and {( ) ( 4) (3 3) (4 )} Q.7 (a) B (b) A (c) D (d) A (e) D Q.8 (a) D ; (b) A Q.9 (a) D (b) A Q.0 C Q. (a) A ; (b) D Part : (A) Only one correct option Eercise-4 log0.3 ( ). The domain of the function f() = is 8 (A) ( 4) (B) ( 4) (C) ( 4) (D) [ ). The function f() = cot ( 3) + cos 3 is defined on the set S where S is equal to: (A) {0 3} (B) (0 3) (C) {0 3} (D) [ 3 0] 3. The range of the function f () = sin + cos where [ ] is the greatest integer function is: (A) (B) 0 (C) { } (D) 0 4. Range of f() = log { 5 (sin cos) + 3} is 3 (A) [0 ] (B) [0 ] (C) 0 (D) none of these ; Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 7 of 4

15 5. Range of f() = is (A) (0 ) (B) ( ) (C) ( ) (D) (3 ) 6. If and y satisfy the equation y = [] + 3 and y = 3 [ ] simultaneously the [ + y] is (A) (B) 9 (C) 30 (D) 7. The function f : [ ) Y defined by f() = is both oneone & onto if (A) Y = R (B) Y = [ ) (C) Y = [4 ) (D) Y = [5 ) 8. Let S be the set of all triangles and R + be the set of positive real numbers. Then the function f : S R + f () = area of the where S is : (A) injective but not surjective (B) surjective but not injective (C) injective as well as surjective (D) neither injective nor surjective 9. Let f() be a function whose domain is [ 5 7]. Let g() = + 5. Then domain of (fog) () is (A) [ 4 ] (B) [ 5 ] (C) [ 6 ] (D) none of these e e 0. The inverse of the function y = is e e (A) log (B) log (C) log (D) log ( + ). The fundamental period of the function f() = + a [ + b] + sin + cos + sin 3 + cos sin (n ) + cos n for every a b R is: (where [ ] denotes the greatest integer function) (A) (B) 4 (C) (D) 0 cos 4 [] cos. The period of e is (where [ ] denotes the greatest integer function) (A) (B) (C) 3 (D) 4 3. If y = f() satisfies the condition f = + ( 0) then f() = (A) (B) (C) (D) a a 4. Given the function f() = (a > 0). If f( + y) + f( y) = k f(). f(y) then k has the value equal to: (A) (B) (C) 4 (D) / 5. A function f : R R satisfies the condition f() + f( ) = 4. Then f() is: (A) (B) + (C) (D) 4 is: 6. The domain of the function f () = cos ( ) tan 3 (A) ( 0) (B) ( 0) (C) ( 0] 6 6 (D) If f () = [] + cos then f: R R is: (where [ ] denotes greatest integer function). (A) oneone and onto (B) oneone and into (C) manyone and into (D) manyone and onto 8. If q 4 p r = 0 p > 0 then the domain of the function f () = log (p 3 + (p + q) + (q + r) + r) is: q q (A) R (B) R ( ] (C) R ( ) (D) none of these q p 9. If [ cos ] + [ sin ] = 3 then the range of the function f () = sin + 3 cos in [0 ] is: (where [. ] denotes greatest integer function) p p (A) [ ) (B) ( ] (C) ( ) (D) [ 3 ) 0. The domain of the function f () = log / log is: 4 (A) 0 < < (B) 0 < (C) (D) null set. The range of the functions f () = log log 6sin is (A) ( ) (B) ( ) (C) ( ] (D) ( ] 3. The domain of the function f () = sin + sin (sin) + log 3/ (3{} + ) ( + ) where {} represents fractional part function is: (A) {} (B) R { } (C) > 3 I (D) none of these 3. The minimum value of f() = a tan + b cot equals the maimum value of g() = a sin + b cos where a > b > 0 when (A) 4a = b (B) 3a = b (C) a = 3b (D) a = 4b 4. Let f ( 4) ( 3) be a function defined by f () = (where [. ] denotes the greatest integer function) then f () is equal to : (A) (B) + (C) + (D) 5. The image of the interval R when the mapping f: R R given by f() = cot ( 4 + 3) is 3 3 (A) (B) 4 4 (C) (0 ) (D) Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 8 of 4

16 a 6. If the graph of the function f () = n is symmetric about y-ais then n is equal to: (a ) (A) (B) / 3 (C) / 4 (D) / 3 7. If f() = cot : R + 0 and g() = : R R. Then the range of the function f(g()) wherever define is (A) 0 (B) 0 (C) 4 (D) Let f: (e ) R be defined by f() =n (n(n )) then (A) f is one one but not onto (B) f is on to but not one - one (C) f is one-one and onto (D) f is neither one-one nor onto 9. Let f: (e ) R be defined by f() =n (n(n )) then (A) f is one one but not onto (B) f is on to but not one - one (C) f is one-one and onto (D) f is neither one-one nor onto 30. Let f() = sin and g() = n if composite functions fog() and gof () are defined and have ranges R & R respectively then. (A) R = {u: < u < } R = {v: 0 < v < } (B) R = {u: < u < 0} R = {v: < v < } (C) R = {u: 0 < u < } R = {v: < v < ; v 0} (D) R = {u: < u < } R = {v:0 < v < } 3. Function f : ( ) (0 e 5 ( 3 ) ] defined by f() = e is (A) many one and onto (B) many one and into (C) one one and onto (D) one one and into 3. The number of solutions of the equation [sin ] = [] where [. ] denotes the greatest integer function is (A) 0 (B) (C) (D) infinitely many 33. The function f() = e + + is (A) an odd function (B) an even function (C) neither an odd nor an even function (D) a periodic function Part : (B) May have more than one options correct 34. For the function f() = n (sin og ) (A) Domain is (B) Range is n (C) Domain is ( ] (D) Range is R 35. A function ' f ' from the set of natural numbers to integers defined by n f (n) = when n is odd n is: when n iseven (A) one-one (B) many-one (C) onto (D) into 36. Domain of f() = sin [ 4 ] where [] denotes greatest integer function is: 3 (A) 3 {0} (B) {0} (C) (D) 37. If F () = sin [] {} then F () is: (A) periodic with fundamental period (B) even (C) range is singleton (D) identical to sgn {} sgn where {} denotes fractional part function and [. ] denotes greatest {} integer function and sgn () is a signum function. 38. D [ ] is the domain of the following functions state which of them are injective. (A) f() = (B) g() = 3 (C) h() = sin (D) k() = sin (/) Eercise-5. Find the domain of the function f() = log ( 0 ) + 3. Find the domain of the function f() = + 3 sin 3. Find the inverse of the following functions. f() = n ( + ) 4. Let f : 3 6 f (). B defined by f () = cos + 5. Find for what values of the following functions would be identical sin +. Find the B such that f eists. Also find Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 9 of 4

17 6. If f() = f () = log ( - ) - log ( - ) and g () = log. 4 then show that f() + f( ) = 4 7. Let f() be a polynomial function satisfying the relation f(). f = f() + f R {0} and f(3) = 6. Determine f(). 8. Find the domain of definitions of the following functions. (i) f () = 3 (ii) f () = (iii) f () = og 0 ( og 0 ( 5 + 6)) + 9. Find the range of the following functions. 4 4 (i) f () = (ii) f () = sin og 4 (iii) f ()= (iv) f () = sin + cos 4 0. Solve the following equation for (where [] & {} denotes integral and fractional part of ) + 3 [] 4 { } = 4. Draw the graph of following functions where [.] denotes greatest integer function and {.} denotes fractional part function. (i) y = {sin } (ii) y = [] + { }. Draw the graph of the function f() = 4 3 and also find the set of values of a for which the equation f() = a has eactly four distinct real roots. 3. Eamine whether the following functions are even or odd or none. 7 ( ) (i) f () = (ii) f () = [ ] [ ] (iii) f () = (sin tan) where [ ] denotes greatest integer function Find the period of the following functions. (i) (ii) sin f () = cot cos tan f () = tan [ ] where [.] denotes greatest integer function. (iii) f () = 5. If f() = sin sin cos cos (iv) f () = sin cos sin3 cos3 and g() = ; < < then define the function fog(). 6. Find the set of real for which the function f () = greatest integer not greater than. [ is not defined where [] denotes the ] [ ] cos sin 4 cos 3 7. Given the functions f() = e g() = cosec & the function 3 h() = f() defined only for those values of which are common to the domains of the functions f() and g(). Calculate the range of the function h(). 8. Let f be a real valued function defined for all real numbers such that for some positive constant a the equation f( a) f() f() holds for all. Prove that the function f is periodic. 9. If f () = g () = 3 Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 30 of 4

18 Then find fog () gof () fof() & gog(). Draw rough sketch of the graphs of fog () & gof (). 0. Find the integral solutions to the equation [] [y] = + y. Show that all the non-integral solutions lie on eactly two lines. Determine these lines. Here [.] denotes greatest integer function. FREE Download Study Package from website: & Eercise-4. D. C 3. C 4. B 5. B 6. C 7. B 8. B 9. C 0. A. A. B 3. D 4. B 5. B 6. D 7. C 8. B 9. D 0. D. D. D 3. D 4. C 5. D 6. D 7. C 8. A 9. C 30. D 3. D 3. B 33. B 34. BC 35. AC 36. B 37. ABCD 38. BD Eercise-5. [ 0) (0 ) f = e e 4. B = [0 4] ; f () = sin 6 5. ( ) (i) [0 ] (ii) (iii) ( 3) 9. (i) 0.. (i) (ii) (ii) [ ] (iii) [4 ) (iv) 4. a ( 3) {0} 3. (i) neither even nor odd (ii) even (iii) odd 4. (i) (ii) (iii) (iv) 5. f(g()) = (0 ) U {... } U ( 3) Period a 9. fog () = ( ) 0 ; 0 gof() = 3 5 fof () = 4 gog() = 4 For 39 Years Que. from IIT-JEE(Advanced) & 5 Years Que. from AIEEE (JEE Main) we distributed a book in class room ; e 6 e 0. Integral solution (0 0); ( ). + y = 6 + y = 0 Teko Classes Maths : Suhag R. Kariya (S. R. K. Sir) Bhopal Phone : page 3 of 4