EXERCISE I 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 log 3 log ( 3 + ) (iii) f () = l n (iv) f () = 7 7 (v) y = log0 sin( 3) 6 (vi) f () = log 0 log00 (vii) f () = ln ( ) (viii) f () = log (i) f( ) 9 http://akj9.wordpress.com (i) f() = log (cos ) (ii) f () = (iii) f() = log log [] (v) f() = log sin / 3 log 00 (vi) f() = log / sin (vii) f () = [ ] + log {} ( 3 + 0) + (viii) f () = (i) If f() = () f () = ( 3 0). ln ( 3) cos 6 3 6 (iv) f() = ( { } ) 3 0 log ( ), [ ] + log log log log log 3 0 0 0 0 0 + sec(sin ) 7 ( 6 ) l n + (7 ) + ln & 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 (iv) f () = (ii) y = (v) y = (iii) f() = 3 (vi) f () = log (cosec - ) ( [sin ] [sin ] ) (vii) f () = 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 ( ) 3 6
Q. Classify the following functions f() definzed in R R as injective, surjective, both or none. (a) f() = 30 8 8 http://akj9.wordpress.com (b) f() = 3 6 + 6 (c) f() = ( + + ) ( + 3) Q. 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. Q.6 If f() = sin² + sin² cos cos and g, then find (gof) (). 3 3 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() = (g) f()= e (h) f() = [(+)²] /3 + [( )²] /3 (ii) If f is an even function defined on the interval (, ), then find the 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) 0 + 0 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. The function 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. Compute the inverse of the functions: (a) f() = ln (b) f() = (c) y = 0 0 0 0
Q.6 A function f :, 3, 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 + cos (b) f() = cos (c) f()= sin+cos (d) f()= cos 3 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 (/n). EXERCISE II Q. Let f be a oneone function with domain {,y,z} 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). (b) Suppose f is a real function satisfying f ( + f ()) = f () and f () =. Find the value of f (). (c) Let 'f' be a function defined from R + R +. If [ f (y)] = (d) f () = 6, find the value of f (0). 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() = 00 and f()+ f()+... + f(n) = n f(n) for all n >. Find the value of f(00). (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 Q. Let f () = for real a, b and c with a 0. If the vertical asymptote of y = f () is = and the c vertical asymptote of y = f () is = 3, find the value(s) that b can take on. http://akj9.wordpress.com
Q. A function f : R R satisfies the condition, f () + f ( ) =. 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 is 0. If r () is the remainder when p() is divided by ( )( ), find the value of r (006). e Q.7 Prove that the function defined as, f () = {} ln{} ln{} {} f () is odd as well as even. ( where {} denotes the fractional part function ) where ever it eists otherwise, then Q.8 In a function f() + f f sin = 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 y = k for + y 0. 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 () = +, 0 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 + =. Q.3 Let {} & [] denote the fractional and integral part of a real number respectively. Solve {}= + [] Q. Let f () = 9 9 3 then find the value of the sum f + f 006 006 3 00 + f +...+ f 006 006 Q. Let f () = ( + )( + )( + 3)( + ) + 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 () = 3 3 Q.7 The set of real values of '' satisfying the equality + = (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. http://akj9.wordpress.com
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 III 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() = (B) f() = sin, g() = (C) f() =, g() = sin (D) f & g cannot be determined (b) If f() = 3, then f () (A) is given by 3 (B) is given by 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. If the function f : [, ) [, ) is defined by f() = ( ), then f () is : [ JEE '99, ] (A) ( ) (B) log (C) log (D) not defined Q. The domain of definition of the function, y () given by the equation, + y = is : (A) 0 < (B) 0 (C) < 0 (D) < < [ JEE 000 Screening), out of 3 ] Q.6 Given = {,, 3, }, find all oneone, onto mappings, f : X X such that, f () =, f () and f (). [ REE 000, 3 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) http://akj9.wordpress.com (B) (C) (D)
log ( 3) (c) The domain of definition of f () = is : 3 (A) R \ {, } (B) (, ) (C) R\{,, 3} (D) ( 3, ) \ {, } (d) (e) Let E = {,, 3, } & F = {, }. Then the number of onto functions from E to F is (A) (B) 6 (C) (D) 8 Let f () =,. Then for what value of is f (f ()) =? (A) (B) (C) (D). [ JEE 00 (Screening) = ] 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 [JEE 00 (Screening), 3 + 3] 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), Q.(a) If the functions f () and g () are defined on R R such that f () = 0,, then (f g)() is (A) one-one and onto (C) one-one but not onto rational, g () = 0, irrational irrational, rational (B) neither one-one nor onto (D) onto but not one-one (D) 0, [JEE 00 (Screening)] (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 00 (Scr.)] http://akj9.wordpress.com
ANSWER KEY FUNCTIONS EXERCISE I Q. (i) 3,, 3, (ii), (, )(iii) (, 3] (iv) (, ) [0, ) (v) (3 < < 3 ) U (3 < ) (vi) 0,, 00 00 0 (vii) ( < < /) U ( > ) (viii) 0,, (i) (3, ] U {0} U [,3 ) () { } [, ) (i) (0, /) U (3/, ) U { : N, } (ii),, 6 6 3 3 (iii) [ 3, ) [ 3,) (iv) (v) K < < (K + ) but where K is nonnegative integer (vi) { 000 < 0000} (vii) (, ) U (, 0) U (, ) (viii) (, ), (i) (, 3) (3, ] [, ) Q. (i) D : R R : [0, ] (ii) D = R ; range [, ] (iii) D : { R ; 3 ; } R : {f()f()r, f() / ; f() } (iv) D : R ; R : (, ) (v) D : R : 3, 6 (vi) D : (n, (n + )) n n n n I,, 6 6, and R : log a ; a (0, ) {} Range is (, ) {0} (vii) D : [, ) {}; R : 0,, 6 6 3 Q. (a) neither surjective nor injective (b) surjective but not injective (c) neither injective nor surjective Q. 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, 3 3 (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 http://akj9.wordpress.com
Q.3 g() if 0 if Q. {, } Q. (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] Q.8 (a) / (b) (c) / (d) 70 Q.0 ±, ± 3, ±, ± Q. f () = y Q. (a) 3/, (b) 6, (c) 30, (d) 0, (e) 00 Q.3 (a), (b), (c) [0, ), (d) 00 EXERCISE II Q. b can be any real number ecept Q. f () =, D = R ; range =(, ] Q.6 606 Q 9. f () = Q. fog () = ( ), 0, 0 ; gof () =, 0 3,, 3, 3 ; fof () =, 0, 3 ; gog () =, 0, 0, 3 3 3 3 3 Q.,, Q.3 = 0 or /3 Q. 00. Q. 09 Q.6 g () = 3 + sin(n + ), n I Q.7 0 Q 8. (0, ) {,,..., } (, 3) Q 9. f () = sin + 3 EXERCISE III 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. B Q. D Q.6 {(, ), (, 3), (3, ), (, )} ; {(, ), (, ), (3, ), (, 3)} and {(, ), (, ), (3, 3), (, )} 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 http://akj9.wordpress.com