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Functions 16 1. ( f + g) x = f ( x ) + g( x ) with domain D1 ∩ D2 Constant...

Functions 16 1. ( f + g) x = f ( x ) + g( x ) with domain D1 ∩ D2 Constant 2. ( f − g) x = f ( x ) − g( x ) with domain D1 ∩ D2 A quantity which retains the same value throughout a 3. ( fg) x = f ( x ) ⋅ g( x ) with domain D1 ∩ D2 set of mathematical operation is called a constant. f f(x) 4. ( x ) = , provided g( x ) ≠ 0 with domain D1 ∩ D2 g g( x ) Variable Example 1. The domain of the function f ( x) = x2 − 4 is A quantity which can take a number of values is called a variable. (a) (−∞, +∞) (b) (−∞, −2] (c) (−∞, −2 ] ∪ [2, ∞) (d) None of these Variables are of two types (i) Independent variable A variable which can Solution (c) Since, f ( x) is real number for which x2 − 4 ≥ 0 take an arbitrary value from a given set is called an ⇒ x2 ≥ 4 ⇒ x ≥ ± 2 independent variable. ⇒ x ∈ ( −∞ , − 2] ∪ [2, ∞) (ii) Dependent variable A variable whose value ∴ Required domain of f is ( −∞ , − 2] ∪ [2, ∞). depends upon the value of independent variable is called a dependent variable. Example 2. The domain and range of the function 1 f ( x) = are, respectively Function 2 − cos 3x 1  1  Let X and Y be two non-empty sets. A function f of X (a) R,  ,1 (b) R − {} 1 ,  ,2 into Y (or from X to Y ) written as f : X → Y is a rule or a 3  3  correspondence which connects every member, say x of X (c) R − {2}, [2, 3] (d) None of these to exactly one member say y of Y. Solution (a) Domain Since, −1 ≤ cos 3x ≤ 1 for all x ∈ R , therefore −1 ≤ − cos 3x ≤ 1 ⇒1 ≤ 2 − cos 3x ≤ 3 for all x ∈ R Domain and Range of a Function ⇒ f ( x) is defined for all x ∈ R. The set X is domain (or definition or inputs) of f and Y is ∴ Domain of f is a set of real number R. called codomain of f. The set of f ( x ), {f ( x ), x ∈ X } consisting 1 Range Let f ( x) = y ⇒ =y of all images f (x), is called the range of f i.e., range is 2 − cos 3x always a subset of the codomain. Range is also set of 1 1 outputs. ⇒ 2 − cos 3x = ⇒ cos 3x = 2 − y y % y = f (x ) is not a function ( y as a function of x), if there is an input 1 for which we get more than one outputs ⇒ −1 ≤ 2 − ≤ 1 [Q − 1 ≤ cos 3x ≤ 1] y % y = f (x ) is not a function, if there is an element in the set of X for 1 which we do not have it’s image (corresponding element in set Y). ⇒ −3 ≤ − ≤ −1 y 1 ⇒ 3 ≥ ≥1 Algebraic Operation on Functions y (Combination of Functions) 1 1  ⇒ ≤ y ≤ 1 ⇒ y ∈  , 1 3 3  If f and g are two functions with domains D1 and D2 1  respectively, then ∴ Range of f is  , 1. 3  298 NDA/NA Mathematics Intervals Onto and Into Functions Closed interval If a real number x is such that If each element in codomain have atleast one a ≤ x ≤ b,then x lies in closed interval a and b and is written pre-image in the set of domain, then the function is onto as x ∈ [a , b]. e.g., if x ∈[2, 3] ⇒ 2 ≤ x ≤ 3 (here end points are and if there is even single element in the set of codomain included). which does not have it’s pre-image, then it is into function Open interval If a real number x is such that or if range is same as codomain, then function is onto and if a < x < b,then x lies in the open interval a and b and written range is proper subset of codomain, then the function is as x ∈( a , b). e.g., x ∈ ( 2,3) ⇒ 2 < x < 3. (here, end points are into. Onto functions are also called ‘surjective functions’. not included). Interval of real number R is written as e.g., ( −∞ , ∞ ).  π π 1. If a function f : − ,  → R is defined as y = sin x. Half open, half closed interval If x is such that  2 2 a < x ≤ b. It is written as ( a , b] or x ∈ ] a , b ] e.g., Here, codomain of this function is R while range is x ∈ ( 2, 3] ⇒ 2 < x ≤ 3 (here, left end point is not included [−1, 1]. Hence, the function is into function. but right end point is included). 2. If a function f: [− 1, 1] → [0, 1] is defined as y = x 2. Half closed, half open interval If x is such that Here, codomain of this function is [0, 1] and range is a ≤ x < b. It is written as [a , b) or [a , b[ e.g., x ∈ [2, 3) or [2, 3[ also [0, 1], hence, the function is onto function. ⇒ 2 ≤ x < 3. (here, left end point is included but right end point is not included.). % Functions which are one-one functions as well as onto functions are called ‘bijective functions’. One-one and many-one functions If f ( x1 ) = f ( x2 ) ⇒ x1 = x2 , then function f ( x ) is one-one. i.e., If x2 distinct elements in domain of a function f have distinct Example 3. If the function f : R → A given by f ( x) = is images in codomain, then f is said to be one-one and if, x2 + 1 there exist atleast two distinct elements in domain having a surjection, then the set A is same image,then it is many-one function. One-one (a) [0, 1) (b) (1,2) functions are also called injective functions. (c) (2,3) (d) None of these y-axis y-axis x2 Solution (a) We have, y = f ( x ) = x2 + 1 ⇒ y ( x2 + 1) = x2 ⇒ x2 (1 − y) = y y y ⇒ x2 = ⇒x= x-axis x-axis 1− y 1− y O O y One-one function Many-one function Since, x is real, therefore ≥ 0 ⇒0 ≤ y 0 for all x in domain or f ′( x ) < 0 for all x in Identity function The function f : R → R is said domain. to be an identity function, iff 4. Any continuous function f(x) which has atleast one f(x) = x, ∀ x ∈ R local maxima or local minima, is many-one. Domain of f = R and Range of f = R 5. If there is any line parallel to x-axis intersecting the Polynomial function A function f : R → R graph of y = f ( x ) in more than one point, then defined by f ( x ) = a0 + a1 x + a2 x 2 +... + an x n , where function is many-one. a0 , a1 , a2 ,... , an are real constants and n is a non-negative 6. We put f ( x1 ) = f ( x2 ). Since, x1 = x2 always satisfies integer, is called a polynomial function. f ( x1 ) = f ( x2 ), so ( x1 − x2 ) will be a factor of f ( x1 ) − f ( x2 ). Rational function Let Hence, we can write as P ( x ) = a0x n + a1x n − 1 + K + an f ( x1 ) − f ( x2 ) = ( x1 − x2 ) g( x1 , x2 ), where g( x1 , x2 ) is some function of x1 and x2. Now, if g( x1 , x2 ) = 0 and Q( x ) = b0x m + b1x m − 1 + K + bm and x1 ≠ x2, hence f ( x ) is many-one. P( x) Then, f ( x ) = is called a rational function. Q( x ) Functions 299 Exponential function The function defined by Properties of Modulus Functions f ( x ) = a x , x ∈ R is called an exponential function. 1.|x|≤ a ⇔ − a ≤ x ≤ a 2.|x|≥ a ⇒ x ≤ − a or x ≥ a Domain of f = R and Range of f = ( 0, ∞ ) 3.|x ± y|≤|x| +| y| 4.|x ± y|≥||x| −| y|| Logarithmic function The function defined by f ( x ) = loga x is called a logarithmic function. Domain of logarithmic function = ( 0, ∞ ), Range of Signum Function logarithmic function = R |x|  1, if x > 0  ,x ≠ 0  Trigonometric function The functions f(x) =  x =  −1, if x < 0 involving circular functions of variable angles are known  0 , x = 0  0, if x = 0  as trigonometric function. y-axis Function Domain Range sin x R [ − 1, 1 ] 1 cos x R [ − 1, 1 ] tan x π R R − {( 2 n + 1) : n ∈ I} x-axis 2 O cot x R − {nπ : n ∈ I} R sec x π ( − ∞, − 1 ] ∪ [1, ∞) –1 R − {(2 n + 1) : n ∈ I} 2 cosec x R − {nπ : n ∈ I} ( − ∞, − 1 ] ∪ [1, ∞) It is written as sign x. Domain ∈ R, Range ∈ {−1, 0, 1}. Inverse trigonometric function The Even and Odd Functions functions involving inverse trigonometric ratios are known as inverse trigonometric functions. A function is said to be an even function, if f ( − x ) = f ( x ), e.g., f ( x ) = cos x , f ( x ) = x 2 etc. Function Domain Range On the other hand a function is said to be an odd −1 sin x [ − 1, 1 ] R function, if cos −1 x [ − 1, 1 ] R f(− x) = − f(x) tan −1 x R π e.g., f ( x ) = sin x , f ( x ) = x3 etc. R − {( 2 n + 1) : n ∈ I} 2 cot−1 x R R − {nπ : n ∈ I} sec −1 x ( − ∞, − 1 ] ∪ [1, ∞) π Properties of Even and Odd R − {( 2 n + 1) : n ∈ I} 2 Functions cosec −1 x ( − ∞ , − 1 ] ∪ [1, ∞) R − {n π : n ∈ I} 1. Derivative of an odd function is an even function and derivative of an even function is an odd function. Absolute Value Function 2. Each function can be expressed as the sum of an even and an odd function. (Modulus Function) 3. Graph of an even function is symmetrical about y-axis  x, x ≥ 0 and the graph of an odd function is symmetric about y =|x| =  − x, x < 0 origin. 4. f ( x ) = 0 is both an even and an odd function. Also, x 2 =|x| 5. f ( x ) = constant, is an even function. y-axis Even Extension and Odd y=–x y=x Extension of a Function Let y = f ( x ) is defined in [a , b], then y = f ( − x ) is it’s even extension in [− b, − a ] and y = − f ( − x ) is it’s odd extension in [− b, − a ]. 45° 45° x-axis % Here, a and b both are either non-negative or non-positive. O 300 NDA/NA Mathematics Example 4. If f is an even function defined on the interval 5. If f ( x ) is a periodic function with period T and g (x) is [ −5, 5] , then the number of real values of x satisfying the a monotonic function, then g (f (x)) is also a periodic  x + 1 function with same period T as that of f (x). equation f ( x) = f   is x 2x  x + 2 e.g., f ( x ) = sin + cos is a periodic function with 3 3 (a) 1 (b) 2 (c) 3 (d) 4 LCM of 6π and 3π, so period of f ( x ) is 6π. Solution (d) Since, f ( x ) is an even function defined on [ −5, 5]. Therefore, f ( − x ) = f ( x ) for all x ∈ [ −5, 5] Example 5. Let f ( x) be a function and k be a positive real  x + 1 number such that f ( x + k) + f ( x) = 0, ∀ x ∈ R, then f ( x) is a Q f ( x) = f   periodic function with period  x + 2 (a) k (b) 2k x+1  x + 1 (c) 3k (d) 4k ⇒ x= or x = −   x+2  x + 2 Solution (b) We have, ⇒ x2 + x − 1 = 0 f ( x + k ) + f ( x ) = 0 , ∀ x ∈R or x + 3x + 1 = 0 2 ⇒ f ( x + k ) = − f ( x ), ∀ x ∈ R −1 ± 5 ⇒ f ( x + 2k) = − f ( x + k), ∀ x ∈ R ⇒ x= 2 ⇒ f ( x + 2k) = f ( x ) −3 ± 5 ∴ f ( x) is a periodic function with period 2k. or x= 2 −1 + 5 −1 − 5 −3 + 5 Hence, the values of x are 2 , 2 , 2 Identical Functions or Equal and −3 − 5. Functions 2 Two functions y = f ( x )and y = g( x )are equal functions, if f ( x ) = g( x ) and they have same domain and range. Periodic Function e.g., f ( x ) = x and g( x ) = x2 x3 and g( x ) = 2 are equal A function f : X → Y is said to be a periodic function, if x x there exist a positive real number T such that functions (identical functions). f ( x + T ) = f ( x ), ∀ x ∈ X. The least value of T is called the fundamental period of a function. In general, the Explicit Functions fundamental period (principal period) is called the period of a function. Graphically, graph gets repeated after each y is said to be an explicit function of the independent period of the function. variable x, if we can write y = f ( x ) i.e., if y can be expressed directly in terms of the independent variable x. e.g., y = x 2 , y = 2x3 + 3x 2 + x + 9 Properties of a Periodic Function 1. If f ( x ) is a periodic function with fundamental period 1 Implicit Functions T, then will also be a periodic function with same f(x) y is said to be an implicit functions of the independent fundamental period T. variable x, if it cannot be expressed directly in terms of x. 2. If f ( x ) is a periodic function with period T, then e.g., x3 y 2 + 3xy 2 + x3 + y3 + 3 y + 2 x + 5 = 0 f ( ax + b) is also a periodic function with fundamental T period |a|. Invertible Functions 3. If f ( x ) is a periodic function with period T, then A function y = f ( x )is said to be invertible if and only if f is one-one onto. af ( x ) + b is also a periodic function with same fundamental period T. All one-one and onto functions (bijective functions) are invertible functions. Their graphs are symmetrical about 4. If f ( x ) and g( x ) are two functions with fundamental the line y = x. periods T1 and T2 respectively, then f ( x ) + g( x ) is a periodic function with fundamental period LCM of T1 and T2 , provided f ( x ) and g( x ) cannot be Method of Finding Inverse of a interchanged by adding a positive number in x Function which is less than LCM of T1 and T2, in that case period becomes that number and also LCM of T1 Suppose, y is written as a function of x. l First find x in terms of y (step 1) and T2 should exist otherwise this is not a periodic −1 l and write it as f ( y) (step 2) function. Functions 301 l Now, replace y by x. (step 3) h = gof  π π e.g., y = sin x , D ∈ − ,  and codomain ∈ [−1,1]  2 2 f g −1 −1 x = sin y =f ( y) (let) (step 2) −1 −1 x f (x) g (x) sin x = f (x) (step 3) % If f −1 (x ) is inverse function of y = f (x ), then D f = R f−1 and D f−1 = R f π Example 7. If f ( x) = sin2 x + sin2  x +   3 ( e x − e − x) Example 6. The inverse of the function f ( x) = +1  π ex + e− x + cos x cos  x +   3 is 1/ 2 5 y  y  and g   = 1, then the value of gof ( x) is (a) (b) log    4 2 −y  2 − y 1 (a) 2 (b) 1 (c) 3 (d) 4 (c) (d) None of these 1− y Solution (b) Given that,  π  π e −ex −x f ( x) = sin 2 x + sin 2  x +  + cos x cos  x +  Solution (b) Let y = f ( x) = +1  3  3 ex + e− x 1  2π   π = 1 − cos 2x + 1 − cos 2x +  + cos 2x +  ⇒ e2x − 1 y − 1 = 2x 2   3  3 e +1 π + cos Applying componendo and dividendo rule, 3  y − 1 + 1 e2x − 1 + e2x + 1 y 2e2x = 2x ⇒ = 1   2π    π  1 y − 1 − 1 e − 1− e −1 y − 2 2x −2 = 2 − cos 2x + cos 2x +  + cos 2x +  +  2    3    3 2 y  y  ⇒ = e2x ⇒ 2 x = log   1 5  π  π  π  2 −y  2 − y = − 2 cos   cos 2x +  + cos 2x +   2 2  3  3  3  1  y  ⇒ x= log   1 5  π  π  2  2 − y = − cos 2x +  + cos 2x +   2 2  3  3  1/ 2  y  5 ⇒ x = log   = , ∀ x ∈R  2 − y 4  5 ∴ gof ( x) = g ( f ( x)) = g   Composite Functions  4   5  Let f : X → Y and g : Y → Z be two functions, we define =1  given g   =1  a function h : X → Z by putting h ( x ) = g ( f ( x )).   4  302 NDA/NA Mathematics Comprehensive Approach log a f ( n ) n If a polynomial satisfies the property a = f ( x) f ( x) + f   = f ( x) f   , then f ( x ) = 1 ± xn. 1 1  x  x log x2n = 2n log |x| , where n ∈ N. 1 1 n Domain of polynomial functions is ( − ∞ , ∞). x + ≥ 2 (If x ∈ R+ ) and x + ≤ − 2 (If x ∈ R− ) x x n Range of odd degree polynomial functions is ( − ∞ , ∞). n Two functions y = f ( x) and y = g( x) can be combined (added, n Even functions are many-one functions. subtracted, multiplied or divided) only in common domains. n Any function can be written as sum of an even function and an n Range of f ( x) = a sin x + b cos x + c is odd function. f ( x) + f ( − x) f ( x) − f ( − x) [c − a2 + b 2 , c + a2 + b 2 ] i.e., f ( x) = + 2 2 n Range of f ( x) = ax2 + bx + c is n Every constant function is an even function and f ( x) = 0is an even  D   D function as well as an odd function.  − 4 a , ∞  , if a > 0 and  − ∞ , − 4 a  , if a < 0   n f ( x) = x function is an identity function. (Df ∈ R and Rf ∈ R) (where D = b 2 − 4 ac) n Constant functions are periodic functions with no fundamental a c LCM of a and c period. n LCM of and = b d HCF of b and d n If f ( x) is inverse function of g( x) , then f ( g( x)) = x or g( f ( x)) = x, i.e., same as f −1( f ( x)) = x or f ( f −1( x)) = x. n LCM of a rational number and an irrational number does not exist. n f ( x) = 0 is the only function which is both even and odd. n The graph of an odd function is symmetric about origin. n If f ( x) is an odd function (or even function), f '( x) is an even function n The graph of an even function is symmetric about y-axis. (or odd function). f ( x) is differentiable of R. n If A and B are two different sets, the number of elements in each of n If A and B are finite having m and n elements respectively, then these is n, then number of one-one onto functions= n!. (a) If gof is one-one ⇒ f is one-one  n P , if n ≥ m (a) Number of one-one function from A to B =  m (b) If gof is onto ⇒ f is onto  0 , if n < m (c) If f and g are one-one onto, then gof will also be one-one n onto. (b) Number of onto functions = Σ ( −1) n − r nC r r m n If f ( x) is any function, then r =1 (a) f ( x) + f ( − x) is an even function. (c) Number of bijection = m! (b) f ( x) − f ( − x) is an odd function. n Number of disjoint elements in A and B are m and n respectively , ax + b then number of mappings from A to B = mn. n If y = f ( x) = , then ( fof ) ( x) = x. x−a n If A = B, then number of mappings from A to B = nn.  Exercise Level I 1. Suppose that g ( x ) = 1 + x and 10. Let g( x ) = x3 − 4x + 6. If f ′ ( x ) = g′ ( x ) and f(1) = 2, f ( g (x)) = 3 + 2 x + x, then f (x) is then what is the value of f ( x )? (NDA 2009 II) (a) 1 + 2 x 2 (b) 2 + x 2 (a) x3 − 4x + 3 (b) x3 − 4x + 6 (c) 1 + x (d) 2 + x (c) x3 − 4x + 1 (d) x3 − 4x + 5 3x + 2 2. The domain of the function log( x 2 − 6 x + 6) is 11. If f ( x ) = , then the value of f −1( x ) is 5x − 3 (a) ( − ∞ , ∞ ) 3x + 2 (b) ( − ∞ , 3 − 3 ) ∪ ( 3 + 3 , ∞ ) (a) 4x − 2 (b) 5x − 3 (c) ( − ∞ , 1] ∪ [ 5, ∞ ) x−1 (d) [0, ∞ ) (c) (d) None of these x+1 3. If n is an integer, then domain of the function sin 2x 12. The period of function f ( x ) = sin4 x + cos4 x is is π  π   π (a) π (b) (c) 2π (d) 3π (a) nπ − , nπ (b) nπ , nπ +  2  2   2 13. The domain of the function (c) [( 2n − 1) π , 2nπ ] (d) [2nπ , ( 2n + 1) π ] f ( x ) = log ( x − 4 + 6 − x ) is 4. The domain of the function f ( x ) = exp ( 5x − 3 − 2 x 2 ) (a) [4, ∞ ) (b) ( − ∞ , 6] is (c) [4, 6] (d) None of these (a) [ 3/ 2, ∞ ) (b) [1, 3/ 2] (c) ( − ∞ , 1] (d) (1, 3/ 2) 14. The range of f ( x ) = cos ( x/ 3) is (a) ( − 1 / 3, 1 / 3) (b) [− 1, 1] 5. If f : R → R, g : R → R and g( x ) = x + 3 and (c) (1 / 3, − 1 / 3) (d) ( − 3, 3) ( fog) ( x ) = ( x + 3)2, then what is the value of f( − 3)? (NDA 2010 I) 15. The domain of the function f ( x ) = sin−1 [log2 ( x/ 2)] is (a) − 9 (b) 0 (c) 9 (d) 3 (a) [ 1, 4] (b) [− 4, 1] 1 (c) [− 1, 4] (d) None of these 6. Given that f ( x ) = x + , then f ( x ) is equal to 2 x x+2 16. The range of the function f ( x ) = is (NDA 2007 II) |x + 2| 2 x2 + 1 x  1 (a) { 0, 1} (b) { − 1, 1} (c) R (d) R − { − 2} (a) + 2 (b)  x +  x x +1  x 2 3 17. If f ( x ) = x + , x ∈ R, then what is the value of 1 1 3 2 (c) x + 4 4 (d) x 2 + 2 x x f −1( x )? (NDA 2010 II) 3x 2 3x 9 7. The inverse f −1( x ) of the function f : [R{ 1}] → [R/{ 1}] (a) + (b) − x+1 2 3 2 4 defined by f ( x ) = is 2x 4 2x 2 x−1 (c) − (d) − 3 9 3 3 x+1 x−1 18. If f ( x ) = x and g( x ) =| x|, then what is the value of (a) (b) x−1 x+1 ( f + g) ( x )? (NDA 2008 I) 1 (a) 0, ∀x ∈ R (b) 2 x,∀x ∈ R (c) (d) x x  2x , for x ≥ 0  0, for x ≥ 0 (c)  (d)  8. If f ( x + 1) = x 2 − 3x + 2 , then f ( x ) is equal to  0, for x < 0  2x , for x < 0 (a) x 2 − 5x − 6 (b) x 2 + 5x − 6 10 x − 10− x 19. The inverse of the function y = is equal to (c) x + 5x + 6 2 (d) x 2 − 5x + 6 10 x + 10− x ax + b 1 9. If y = f ( x ) = , then f ( y ) is equal to (a) log10 ( 2 − x ) (b) log10 ( 2x − 1) cx − a 2 (a) 2x (b) x 1 2x 1 1+ x (c) −x (d) −2 x (c) log10 (d) log10 4 2− x 2 1− x 304 NDA/NA Mathematics  1 + x  2x  2x − 1 20. If f ( x ) = log   , then f   is equal to 27. If f ( x ) = , ( x ≠ 5), then f −1( x ) is equal to  1 − x  1 + x2  x+5 x+5 1 5x + 1  1 − x (a) ,x ≠ (b) ,x ≠2 (a) f   (b) f ( x )2 2x − 1 2 2− x  1 + x x−5 1 5x − 1 (c) ,x ≠ − (d) ,x ≠2 (c) 1 (d) 2f ( x ) 2x + 1 2 2− x  1 − x 2x + 1 21. If f ( x ) = log  . What is the value of f ( a ) + f ( b) ? 28. If f ( x ) = , then ( fof ) (2) is equal to  1 + x 3x − 2  a − b (a) 1 (b) 3 (a) f ( a + b) (b) f   (c) 4 (d) 2  1 + ab 1+ x  a + b 29. If f ( x ) = log , then f ( x ) is (c) f   (d) f ( a − b) 1− x  1 + ab (a) even function 22. Let fog ( x ) = F ( x ). If F ( x ) = a 2 − x 2 and g( x ) = − x 2 , (b) f ( x1 ) f ( x2 ) = f ( x1 + x2 ) f ( x1 ) what is the value of f ( x ) ? (c) = f ( x1 − x2 ) f ( x2 ) (a) a2 − x (b) a2 + x (d) odd function (c) a2 + x2 (d) a2 − x2 30. What is the period of the function 1  π f ( x ) =|sin x| +|cos x| ? 23. If g( x ) = sin x, x ∈ R and f ( x ) = , x ∈  0,  , what (a) π/ 2 (b) π sin x  2 (c) 2π (d) π/ 4 is the value of ( gof ) ( x )? (NDA 2008 I) 1 31. What is the inverse of f ( x ) = (1 − x3 )1/ 5 + 2 ? (a) 1 (b) sin (sin x ) (a) [1 − ( x − 2)5 ]1/ 3 (b) [1 + ( x − 2)5 ]1/ 3 1  1  (c) ( x + 2)1/ 3 (d) [1 + ( x + 2)5 ]1/ 3 (c) (d) sin   2 sin ( x )  sin x  32. A function f(x) satisfies the condition  1 1 1 24. If f ( x ) = 2 + , then which one of the following is f  x +  = x + 2 , x ≠ 0. What is the value of f ( x ) ? 2  x x x correct? (a) x 2 − 2 for all x ≠ 0 (a) As x increases, f ( x ) also increases. (b) x 2 − 2 for all x satisfying|x|> 2 (b) As x becomes larger and larger, f ( x ) assumes (c) x 2 − 2 for all x satisfying|x|< 2 values nearer to 2. (d) x 2 + 2 for all x ≠ 0 (c) As x increases, f ( x ) takes values nearer and nearer to 3 for sufficiently large values of x. 3x + 2  3 33. If f ( x ) =  x ≠  , then which one of the (d) As x becomes larger and larger, it is not possible 5x − 3  5 to find the value of f ( x ). following is correct? 1− x  1 − x (a) f −1( x ) = f ( x ) (b) f −1( x ) = − f ( x ) 25. If f ( x ) = , then f   is equal to 1 1+ x  1 + x (c) ( fof ) ( x ) = − x (d) f −1x = − f(x) 1− x 19 (a) x (b) x( x − p) x ( x − q ) 1+ x 34. f ( x ) = + , p ≠ q. What is the value of 1+ x 1 q− p p− q (c) (d) 1− x x f ( p) + f ( q ) ? (a) f ( p − q ) (b) f ( p + q ) 26. Let y 2 = 4ax , a ≠ 0. Now, consider the following (c) f ( p ( p + q )) (d) f ( q ( p − q )) statements  1  I. y = 2 ax expresses y as a function of x. 35. If 2 f ( x + 1) + f   = 2 x and x ≠ − 1, then f ( 2) is  x + 1 II. y = − 2 ax expresses y as a function of x. equal to III. y = ± 2 ax expresses y as a function of x. (a) −1 (b) 2 Which of these is/are correct? (c) 5/3 (d) 5/2 (a) I and II (b) I and III 36. The function f : R → R defined by f ( x ) = ex is (c) II and III (a) onto (b) many-one (d) Only III (c) one-one and into (d) many-one and onto Functions 305 37. If R denotes the set of all real number, then the 44. What is the range of function f ( x ) = ( 3 − cos 2 x )−1 ? function f : R → R defined f ( x ) =| x| is 1   1 1 (a) one-one only (a)  , 1 (b)  ,  4   4 2 (b) onto only  1 1  1 1 (c) Both one-one and onto (c)  − , −  (d)  4 , 2 (d) Neither one-one nor onto  4 2 38. Let f : R → R be defined by f ( x ) = 2 x +| x|, then 45. If X = { 1, 2, 3} and Y = { 0, 1} and f : X → Y defined by f ( 2x ) + f ( − x ) − f ( x ) is equal to f = {(1, 1), ( 2, 1), ( 3, 0)}, then f is (a) 2 x (b) 2|x| (a) one-to-one but not onto (c) − 2x (d) − 2|x| (b) onto but not one-to-one (c) one-to-one and onto 39. If f : R → R and g : R → R defined by f ( x ) = 2 x + 3 and (d) neither one-to-one nor onto g( x ) = x 2 + 7, then the value of x for which 46. Domain of the function f ( x ) = sin−1 (log2 x ) in the set f ( g ( x )) = 25, are (a) ± 1 (b) ± 2 of real numbers is (c) ± 3 (d) ± 4 (a) { x : 1 ≤ x ≤ 2} (b) { x : 1 ≤ x ≤ 3} cos2 x + sin4 x  1  40. If f ( x ) = for x ∈ R, then f( 2002) equals (c) { x : − 1 ≤ x ≤ 2 } (d)  x : ≤ x ≤ 2 sin2 x + cos4 x  2  to 47. If f ( x ) = cos (loge x ) , then (a) 1 (b) 2 1   y  f(x) f( y) − f   + f ( xy ) has the value (c) 3 (d) 4 2   x   1 1 41. If f ( x ) = + , for x > 2, (a) 1 (b) 1/2 x + 2 2x − 4 x − 2 2x − 4 (c) –2 (d) 0 then f(11) is equal to 48. The range of the function sin( sin−1 x + cos−1 x ),| x|≤ 1 (a)7/6 (b) 5/6 is (c) 6/7 (d) 5/7 (a) [− 1, 1] (b) [1, − 1] 42. Let f : R → R : f ( x ) = x 2 and g : R → R: g( x ) = x + 5, (c) { 0} (d) { 1} then gof is 49. The domain of sin−1 (log3 x ) is (a) ( x + 5) (b) ( x + 52 ) (a) [−1, 1] (b) [0, 1] (c) ( x + 5) 2 (d) ( x + 5)2 1  (c) [0, ∞ ] (d)  , 3 43. The function f satisfies the functional equation 3   x + 59 αx 3 f(x) + 2 f   = 10x + 30 for all real x ≠ 1. The 50. If f ( x ) =, x ≠ − 1 for what value of α is f ( f ( x )) = x?  x−1  x+1 value of f( 7) is (a) 2 (b) − 2 (a) 8 (b) 4 (c) −8 (d) 11 (c) −1 (d) 2 Level II 1. The largest possible set of real numbers which can be 4. Domain of f ( x ) = log|log x| is 1 (a) ( 0, ∞ ) (b) (1, ∞ ) the domain of f ( x ) = 1 − is x (c) ( 0, 1) ∪ (1, ∞ ) (d) ( −∞ , 1) (a) ( 0, 1) ∪ ( 0, ∞ ) (b) ( −1, 0) ∪ (1, ∞ ) 5. Which one of the following is a bijective function on (c) ( −∞ , − 1) ∪ ( 0 , ∞ ) (d) ( −∞ , 0) ∪ (1, ∞ ) the set of real numbers? 1 (a) 2x − 5 (b) | x| (c) x 2 (d) x 2 + 1 2. The domain of the function f ( x ) = log is |sin x| 6. If p, q and r are positive integers, ω is the cube root of (a) R − { 2nπ, n ∈ I } (b) R − { nπ, n ∈ I } unity and f ( x ) = x3 p + x3 q + 1 + x3 r + 2, then what is the (c) R − { − π, π } (d) ( − ∞ , ∞ ) value of f(w)? (NDA 2011 II) 10 x − 10− x 3. The inverse of the function is (a) ω (b) − ω 2 (c) − ω (d) 0 10 x + 10− x 1  1 + x 1  1 − x 7. The function y = f ( x ) = mx + c has (NDA 2011 II) (a) log10   (b) log10   2  1 − x 2  1 + x (a) maximum point but no minimum point (b) minimum point but no maximum point 1  2x  (c) log10   (d) None of these (c) both maximum and minimum points 4  2 − x (d) neither maximum point nor minimum point 306 NDA/NA Mathematics 8. Let X and Y be subsets of R, the set of all real 1 17. The domain of the function x + 1 is numbers. The function F : X → Y defined by f ( x ) = x 2 x for x ∈ X is one-one but not onto  3 − 1  5 − 1 (here R + is the set of all positive real numbers). (a) 0 < x <   (b) 0 < x ≤   (a) X = Y = R + (b) X = R , Y = R +  2   2  + (c) X = R , Y = R (d) X = Y = R  −1 − 5   5 − 1 (c)   < x< 0 (d) 0 ≤ x <    2   2  9. The function f :R→ R is defined by f ( x ) = cos2 x + sin4 x for x ∈ R, then f ( R ) equals to 18. If f ( x ) = 2 x + 7 and g( x ) = x 2 + 7 , x ∈ R, then which 3  3  3  3  value of x will satisfy fog( x ) = 25? (NDA 2010 II) (a)  ,1 (b)  , 1 (c)  , 1 (d)  , 1 4  4  4  4  (a) − 1 and 1 (b) − 2 and 2 10 + x (c) − 2 and 2 (d) None of these 10. If e f ( x ) = , x ∈( − 10, 10) and 10 − x 19. Let f : R → R be defined as f ( x ) = x|x|.  200x  f ( x ) = kf   , then k is equal to Which one of the following is correct?  100 + x 2  (a) f is only into. (a) 0.5 (b) 0.6 (b) f is only one-one. (c) 0.7 (d) 0.8 (c) f is neither onto nor one-one. (d) f is one-one and onto. 11. If f : R → R and g : R → R are defined by f ( x ) = 2x + 3 and g( x ) = x 2 + 7, then the values of x such that 20. Let f ( x ) be a differentiable even function. Consider the following statements g( f ( x )) = 8, are I. f ′( x ) is an even function. (a) 1, 2 (b) − 1, 2 II. f ′( x ) is an odd function. (c) − 1, − 2 (d) 1, − 2 III. f ′( x ) may be even or odd. 12. Suppose f : [2, 2] → R is defined by Which of the above statements is/are correct?  −1 for − 2 ≤ x ≤ 0 (a) Only I (b) Only II f (x) =  , (c) I and III (d) II and III  x − 1 for 0 ≤ x ≤ 2 then { x ∈{ −2, 2}: x ≤ 0 and f (| x|) = x} is equal to 21. Let g( x ) be the inverse of an invertible function f ( x ) (a) { − 1} (b) { 0} which is differentiable at x = c. Which one of the (c) { − 1 / 2} (d) φ following is equal to g ′ [ f ( c)]? 1 13. If f ( xy ) = f ( x ) f ( y ), then f(t) may be of the form (a) f ′( c) (b) f ′( c) (NDA 2012 I) 1 (a) t + k (b) ct + k (c) t + c k (d) t k (c) f ( c) (d) f ( c) where R is constant. 22. Let f : R → R be defined by f ( x ) =| x|/ x, x ≠ 0, f( 0) = 2. 14. If the roots of the equation x 2 − 4x − log3 N = 0 are What is the range of f ? (NDA 2009 II) real, then what is the minimum value of N? (a) { 1, 2} (b) { 1, − 1} (c) { −1, 1, 2} (d) { 1} (NDA 2011 II) 23. A function f : R → R is defined by f ( x ) = ( x − 1) ( x − 2). 1 1 1 1 Which one of the following is correct in respect of (a) (b) (c) (d) 256 27 64 81 function? a (a) It is one-one but not onto. 15. If sin θ = x + , ∀ x ∈ R − { 0}, then which one of the (b) It is onto but not one-one. x (c) It is both one-one and onto. following is correct? (NDA 2011 II) (d) It is neither one-one nor onto. 1 (a) a ≥ 4 (b) a ≥ 24. What is period of the function 2 1 1 f ( x ) =|sin x + cos x| +|sin x − cos x| ? (c) a ≤ (d) a ≤ (a) π/ 6 (b) π/ 4 (c) π/ 2 (d) π 4 2 25. Which one of the following is correct for the graph of 16. If a real valued function of a real variable is defined y =|x|? as f ( x ) = cos−1( 3x − 1), then the domain of the (a) It lies only in the first quadrant of the xy-plane. function f, is given by (b) It lies only in the first and third quadrants of the  1 xy-plane. (a)  x : 0 ≤ x ≤  (b) { x : 1 ≤ x ≤ 2}  3 (c) It lies only in the first and second quadrants of the xy-plane.  2  3 (c)  x : 0 ≤ x ≤  (d)  x : 0 ≤ x ≤  (d) It lies only in the third and fourth quadrants of  3   2 the xy-plane. Functions 307 26. If f ( x ) = l x 2 + m x + n , x ∈ R, then what is the value of sec4x + cosec4x 32. Assertion (A) The function f ( x ) = is f ( x + 3) − f ( x ) x3 + x 4 cot x ? f ( x + 2) − f ( x + 1) an odd function. (a) 0 (b) 3 Reason (R) f ( x ) is an odd function, since, 3l − m + n l+m+n f ( x ) = − f ( − x ). (c) (d) l − 2m + n l+m−n 33. Assertion (A) If f ( x ) is a periodic function, then 27. If f ( x ) = ax + b and g( x ) = cx + d such that T f ( ax + b) is a periodic function with period. fog( x ) = gof ( x ), then which one of the following is |a| correct? T Reason (R) Since, f ( ax − b) has a period. (a) f ( a ) = g( c) (b) f ( b) = g( a ) |a| (c) f ( c) = g( d ) (d) f ( d ) = g( b) 34. Assertion (A) The modulus function f ( x ) =|x| is not 28. A rectangular box with a cover is to have a square one-one. base. The volume is to be 10 cu cm. The surface area Reason (R) The negative real numbers are not the of the box in terms of the side x is given by which one images of any real numbers. of the following functions? (NDA 2008 II) x2 (a) f ( x ) = ( 40 / x ) + 2 x 2 35. Assertion (A) If f ( x ) = x and F ( x ) = , then x (b) f ( x ) = ( 40 / x ) + x 2 F ( x ) = f ( x ) always. (c) f ( x ) = ( 40 / x ) + x Reason (R) At x = 0, F ( x ) is not defined. (d) f ( x ) = ( 60 / x ) + 2 x 36. Assertion (A) The function f ( x ) =|x|/ x does not 29. Which one of the following real valued function is pass through the origin. never zero? (NDA 2008 I) Reason (R) f ( x ) is undefined for x ≤ 0. (a) Polynomial function (b) Trigonometric function 37. Match List I with List II and select the correct (c) Logarithmic function answer using the code given below the lists (d) Exponential function List I List II 30. Let f : [− 100 π , 1000 π ] → [− 1, 1] be defined by A. f ( x) = cos x 1. The graph cuts y-axis in f(θ ) = sin θ. Then, what is the number of values of infinite number of points θ ∈ [− 100 π , 1000 π ] for which f(θ ) = 0? (NDA 2007 I) B. f ( x) = ln x 2. The graph cuts x-axis in (a) 1000 (b) 1101 two points (c) 1100 (d) 1110 C. f ( x) = x2 − 5 x + 4 3. The graph cuts y-axis in 31. Let f : R → R be defined as f ( x ) = ax 2 + bx + c, a , b only one point and c being fixed non-zero real numbers. Which one D. f ( x) = e x 4. The graph cuts x-axis in only one point of the following statements is correct in general? (NDA 2007 I) 5. The graph cuts x-axis in −1 infinite number of points (a) If b − 4ac > 0, then f ( 0) does not contain 0 2 (b) If b2 − 4ac < 0, then f −1( 0) must contain 0 (NDA 2007 I) (c) If b2 − 4ac > 0, then f −1( 0) may contain 0 Codes (d) If b2 − 4ac < 0, then f −1( 0) may contain 0 A B C D A B C D (a) 1 4 5 3 (b) 1 3 5 4 (c) 5 4 2 3 (d) 5 3 2 4 Directions (Q. Nos. 32-36) Each of these questions contain two statements, one is Assertion (A) 1 38. Function ‘f ’ defined by f ( x ) = x + , then consider the and other is Reason (R). Each of these questions also has x four alternative choices, only one of which is the correct following statements (NDA 2008 II) answer. You have to select one of the codes (a),(b),(c) and I. { f ( x )} 2 = f ( x 2 ) + 2 (d) given below. II. { f ( x )}3 = f ( x3 ) + 3 f ( x ) Codes Which of the correct in above statements? (a) Both A and R are individually true and R is the (a) Only I (b) Only II correct explanation of A. (c) Both I and II (d) Neither I nor II (b) Both A and R are individually true but R is not 39. Consider the following statements the correct explanation of A. I. Every function has a primitive. (c) A is true but R is false. II. A primitive of a function is unique. (d) A is false but R is true. 308 NDA/NA Mathematics Which of the statements given above is/are correct? 42. The value of f + g, when x < 3 is (NDA 2010 I) (a) x 2 − 3x (b) x − 2 (a) Only I (b) Only II (c) x − 1 (d) None of these (c) Both I and II (d) Neither I nor II f 43. The value of when x ≥ 4 is 40. Consider the following statements g 1− x x−4 I. f ( x ) = is inverse of itself. (a) 1+ x x+2 II. f ( x ) = 5log x is inverse of itself. x−4 (b) 2 Which of the statements given above is/are correct? x + 2x + 2 (a) Only I (b) Only II x+1 (c) (c) Both I and II (d) Neither I nor II x−1 41. Consider the following statements (d) None of these I. The function log ( x + ( x 2 + 1)) is an odd 44. The value of f ( g( x )) when x < 4 is function. (a) ( x − 6)( x − 4) (b) ( x − 6)( x − 2) II. The function cos (log ( x + x 2 + 1 )) is an even (c) ( x − 3) (d) None of these function. III. If f ( x ) is an even function, then curve y = f ( x ) is Directions (Q. Nos. 45-46) Consider the function symmetric about x-axis. f(x) = x − x2 Which of the statements given above is/are correct? (a) Only I (b) Only II 45. The domain of f ( x ) is (c) Only III (d) All I, II and III (a) [0, 2] (b) [0, 1] Directions (Q. Nos. 42-44) Consider (c) [0, 3] (d) None of these  x 2 − 4x + 3, x < 3 46. The range of f ( x ) is f(x) =  ; x − 4 , x ≥ 3  1 (a) [0, 3] (b) 0,  x − 3 ; x < 4  2 g( x ) =  2 (c) [0, 1] (d) None of these  x + 2x + 2, x ≥ 4 , then Answers Level I 1. (b) 2. (c) 3. (b) 4. (b) 5. (c) 6. (b) 7. (a) 8. (d) 9. (b) 10. (d) 11. (b) 12. (b) 13. (c) 14. (b) 15. (a) 16. (b) 17. (b) 18. (c) 19. (d) 20. (d) 21. (c) 22. (b) 23. (d) 24. (b) 25. (a) 26. (d) 27. (b) 28. (d) 29. (d) 30. (a) 31. (a) 32. (a) 33. (a) 34. (b) 35. (c) 36. (c) 37. (d) 38. (b) 39. (b) 40. (a) 41. (c) 42. (c) 43. (b) 44. (d) 45. (b) 46. (d) 47. (d) 48. (d) 49. (d) 50. (c) Level II 1. (d) 2. (b) 3. (a) 4. (c) 5. (a) 6. (d) 7. (d) 8. (d) 9. (c) 10. (a) 11. (c) 12. (c) 13. (d) 14. (d) 15. (c) 16. (c) 17. (b) 18. (c) 19. (a) 20. (b) 21. (b) 22. (c) 23. (b) 24. (c) 25. (c) 26. (b) 27. (d) 28. (a) 29. (d) 30. (c) 31. (a) 32. (a) 33. (b) 34. (a) 35. (d) 36. (c) 37. (c) 38. (c) 39. (b) 40. (a) 41. (d) 42. (a) 43. (b) 44. (a) 45. (b) 46. (b)  Hints & Solutions Level I 1. Given that, g (x) = 1 + x and f ( g (x)) = 3 + 2 x + x... (i) ⇒ f (x) = (x − 1)2 − 3(x − 1) + 2 ⇒ f (1 + x ) = 3 + 2 x + x f (x) = x2 − 2x + 1 − 3x + 3 + 2 Put 1 + x = y ⇒ x = y − 1 ⇒ x = ( y − 1)2 f (x) = x2 − 5 x + 6 Then, f ( y) = 3 + 2( y − 1) + ( y − 1)2 ax + b 9. y = f (x) = ⇒ f ( y) = f { f (x)} cx − a = 3 + 2 y − 2 + y2 + 1 − 2 y = 2 + y 2  ax + b Replacing y on x, we get f (x) = 2 + x2 a +b  cx − a  a 2x + ab + bcx − ab 2. Let the function be f (x) = log (x2 − 6x + 6 ) is defined, = =  ax + b acx + bc − acx + a 2 when log (x2 − 6x + 6) ≥ 0 c  −a  cx − a  ⇒ x2 − 6x + 6 ≥ 1 ⇒ x2 − 6x + 5 ≥ 0 x (a 2 + bc) ⇒ (x − 5) (x − 1) ≥ 0 ⇒ f ( y) = =x (a 2 + bc) This inequality hold, if x ≤ 1 or x ≥ 5. Hence, the domain of the function will be (− ∞ , 1] ∪ [5, ∞ ). 10. Given, f ′ (x) = g′ (x) 3. Let f (x) = sin 2x On integrating both sides, we get As sin 2x can’t be negative. f (x) = g (x) + C We know sin x is positive, if 0 ≤ x ≤ π ⇒ f (x) = x3 − 4x + 6 + C …(i) π Q f (1) = 2 (given) ⇒ sin 2x is positive, if 0 ≤ x ≤ ∴ 2 = 1 − 4 + 6 + C ⇒ C = −1 2 [from Eq. (i)] ∴ Domain of function sin 2x is [nπ , nπ + π /2]. ∴ f (x) = x3 − 4x + 5 4. Given that, f (x) = exp ( 5x − 3 − 2x2 ) 3x + 2 11. y= 5 x − 3 − 2x 2 5x − 3 or f (x) = e ⇒ 5xy − 3 y = 3x + 2 ⇒ (5 y − 3) x = 3 y + 2 For domain, 5x − 3 − 2x2 > 0 3y + 2 ⇒ x= 5y −3 ⇒ 2 x2 − 5 x + 3 ≤ 0 ⇒ 2 x2 − 2 x − 3 x + 3 ≤ 0 3x + 2 3x + 2 ⇒ 2x(x − 1) − 3 (x − 1) ≤ 0 ⇒ (2x − 3) (x − 1) ≤ 0 ⇒ f −1( y) = ⇒ f −1 (x) = 5y −3 5x − 3 3 3 ⇒ x≤ and x ≥ 1 ⇒ 1 ≤ x ≤ 12. Qsin 4 x + cos 4 x = (sin 2 + cos 2 x)2 − 2 sin 2 x cos 2 x 2 2 1 1  1 − cos 4 x 3 1  3 = 1 − (sin 2x )2 = 1 −   = + cos 4x ∴ Domain of f (x) = 1,  2 2 2  4 4  2 Since, cos x is periodic with period 2π, then cos 4x is 5. Q fog (x) = (x + 3)2 = f ( g (x)) 2π π periodic with period =. and g (x) = x + 3 4 2 ⇒ f (x + 3) = (x + 3)2 π So, the period of function f (x) = sin 4 x + cos 4 x is. ⇒ f (x) = x2 2 ⇒ f (−3) = (−3)2 = 9 13. Given that, f (x) = log ( x − 4 + 6 − x ) 2 ⇒ x − 4 ≥ 0 and 6 − x ≥ 0 ⇒ x ≥ 4 and x ≤ 6  1  1 6. Given that, f (x) =  x + ⇒ f (x) =  x + 2  ∴ Domain of f (x) = [4, 6 ]  x  x 14. Given that, f (x) = cos (x/3) x+1 x+1 7. Let y = f (x) =. Now, y = We know that − 1 ≤ cos (x/3) ≤ 1 x−1 x−1 ∴ Range of f (x) = [−1, 1] ⇒ xy − y = x + 1 ⇒ x ( y − 1) = y + 1 y+1 y+1 15. Given that, f (x) = sin −1 [log 2 (x/2)] ⇒ x= ⇒ f −1 ( y) = Domain of sin −1 x is x ∈ [− 1, 1] y−1 y−1  x x −1 x+1 ⇒ −1 ≤ log 2  ≤ 1 ⇒ 2−1 ≤ ≤ 21 f (x) = (replacing y by x)  2 2 x−1 1 x ⇒ ≤ ≤ 2 ⇒ 1 ≤ x ≤4 8. We have f (x + 1) = x2 − 3x + 2 2 2 Put x = x − 1 ∴ x ∈[1, 4] 310 NDA/NA Mathematics x+2  (1 + ab) − (a + b)  16. Given that, f (x) = = log   | x + 2|  (1 + ab) + (a + b) Now, we can define the function [Q log a + log b = log ab] − 1 , x < − 2  a+b f (x) =   1 −   1, x > − 2 1 + ab   a + b = log  =f   ∴ Range of f (x) is { − 1, 1} 1 + a + b   1 + ab   2 3  1 + ab  17. f (x) = x + = y (say) 3 2 22. Given that, F (x) = a 2 − x2 and g (x) = − x2 ⇒ 4x + 9 = 6 y 6y − 9 Let f (x) = a 2 + x ⇒ x= = f −1 ( y) 4 then ( fog )(x) = f [ g (x)] 6x − 9 3x 9 ⇒ a 2 − x2 = F (x) ⇒ f −1(x) = = − 4 2 4 1 18. Q f (x) = x and g (x) = | x| 23. Q g (x) = sin x and f (x) = sin x ∴ ( f + g ) (x) = f (x) + g (x)  1  = x + | x| ( gof )(x) = g [ f (x)] = sin f (x) = sin    sin x x + x, x ≥ 0 = (by redefining the function) 1  x − x, x < 0 24. If f (x) = 2 + x 2x, x≥0 = Then, as x becomes larger and larger, f (x) assumes  0, x 0 (3 − cos 2 x) 49. For domain of sin −1 (log3 x), − 1 ≤ log3 x ≤ 1 1 3y − 1 ⇒ 3 − cos 2x = ⇒ cos 2x = ⇒ 3 –1 ≤ x ≤ 3 y y 1  ∴ Domain of sin −1 (log3 x) =  , 3 1 −1  3 y − 1  3  ⇒ x = cos   2  y  αx 50. f (x) = Now, x ∈ R, if x+1 3y − 1 −1 ≤ ≤1  αx y α   αx  x + 1 α 2x ⇒ 1 −1 ≤ 3 − ≤ 1 ∴ f ( f (x)) = f  = = y  x + 1  α x  α x+ x+1   +1 −1 1