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Cengage Maths Solutions Class 12 Relations and Functions - Calculus

Cengage Maths Solutions Class 12 Relations and Functions (https://doubtnut.com/cengage-solutions/chapter-relations-and-functions-1) PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on Relations and Functions and its various sub-topics for effective exam preparation. The Cengage 12 PDF solutions help the student to swiftly practice and revise the important concepts just before the annual exam to score high marks in maths.<br>For more such video solutions/Pdf solution, visit https://doubtnut.com and clear your doubt instantly.

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Cengage Maths Solutions Class 12 Relations and Functions - Calculus

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER CALCULUS RELATIONS AND FUNCTIONS ||  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number 1 Solve (2x + 1)(x - 3)(x + 7) < 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Solve 2 2 x< 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Solve 2x - 3 3 ≥ 3 3x - 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Solve (x - 1)2(x + 4) < 0 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Solve x >√(1 - x) 5  Watch Free Video Solution on Doubtnut

  2. CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number √ √ 1 -√1 - x2 Find the domain of f(x) = 6 1 -  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Solve (x - 1)|x + 1|cosx > 0, f or x ∈ [ - π, π] (x − 1)|x + 1|cosx > 0,f or x 7 ∈ [ − π,π]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Find the domain of f(x) =√x - 4 - 2√(x - 5) -√x - 4 + 2√(x - 5) f(x) = √x − 4 − 2√(x − 5) 8 − √x − 4 + 2√(x − 5)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number x - 3 Find the domain of the following functions: f(x) = 9 (x + 3)√x2- 4  Watch Free Video Solution on Doubtnut

  3. CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number 1 Find the domain of the following functions: f(x) =√2 - x - 10 √9 - x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number x - 2 x + 2+ 1 - x 1 + x √ √ Find the domain of the following functions: f(x) = 11  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number √( ) 2 1 2x - 1 x3+ 1 Find the domain of the following functions: f(x) = - x + 1- x2- x + 1 f(x) 12 2 1 2x − 1 = √( ) − − x2− x + 1 x3+ 1 x + 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number √ x -√1 - x2 Find the domain of the following functions: f(x) = 13

  4.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number x2+ 1 x2+ 2 Find the range of f(x) = 14  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS System And Inequalities AND FUNCTIONS_Number Solve: x ex- 1 (x + 2)(x - 3)2≤ 0. ( ) 15  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain and range of f(x) =√x2- 3x + 2 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of the function f(x) = 6x+ 3x+ 6-x+ 3- x+ 2. 17  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions

  5. 18 Find the domain and range of f(x) =√x2- 4x + 6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of f(x)x2- x + 1 19 x2+ x + 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the complete set of values of a such that x2- x 20 1 - ax attains all real values.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions x2+ 34x - 71 x2+ 2x - 7 21 Find the range of f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of f(x)√x - 1 +√5 - x 22  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions If f(x) =√x2+ ax + 4 is defined for all x, then find the value of a. 23  Watch Free Video Solution on Doubtnut

  6. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain and range of f(x) =√3 - 2x - x2 24  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 2 25 Solve |3x - 2| ≤  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 26 Solve ||x - 1| - 5| ≤ 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 - |x| |x| - 2 √ find the domain of f(x) = 27  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve |x + 3| + x 28 > 1 x + 2  Watch Free Video Solution on Doubtnut

  7. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 29 Solve: |x - 1| + |x - 2| ≥ 4.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve |sinx + cosx| = |sinx| + |cosx|,x ∈ [0, 2π]. |sinx + cosx| = |sinx| + |cosx|,x 30 ∈ [0,2π].  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve: - 2x2+ 1 + ex+ sinx = 2x2- 1 + ex+ |sinx|,x ∈ [0, 2π]. | | | | ∣∣− 2x2+ 1 + ex+ sinx∣∣= ∣∣2x2− 1∣∣+ ex 31 + |sinx|,x ∈ [0,2π].  Watch Free Video Solution on Doubtnut

  8. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 32 Find the set of real value(s) of a for which the equation |2x + 3| + |2x - 3| = ax + 6 has more than two solutions.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 33 If a < b < c, then find the range of f(x) = |x - a| + |x - b| + |x - c|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions √ 1 -√x2- 6x + 9 Find the range of f(x) = 34  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 Find the domain of the function f(x) = 35 1 + 2sinx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 2 1 Solve sinx ≻ or find the domain of f(x) = 36 √1 + 2sinx  Watch Free Video Solution on Doubtnut

  9. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions x 37 Find the number of solutions of sinx = 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the number of solutions of the equation sinx = x2+ x + 1. 38  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of f(x) = sin2x - sinx + 1. 39  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 Find the range of f(x) = 40 2cosx - 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions sinx cosx if:f(x) = , then find the range of f(x) - 41 √1 + tan2x √1 + cot2x  Watch Free Video Solution on Doubtnut

  10. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 42 Find the range of f(x) = |sinx| + |cosx|,x ∈ R.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the range of f(θ) = 5cosθ + 3cos θ +π 43 + 3 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain of f(x) =√sinx +√16 - x2 44  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve (a) tanx < 2(b)cosx ≤ -1 45 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) π 4,π . Prove that the least positive value of x, satisfying tanx = x + 1, liesinthe ∈ terval 2 46

  11. π π tanx = x + 1,liesinthe ∈ terval( ) , 4 2 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( π 4cos2x , where `-oo ) 47 Find the range of f(x) = sec  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 48 If x ∈ [1, 2] , then find the range of f(x) = tanx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 Find the range of f(x) = 49 1 - 3√1 - sin2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 2sin2x + 2sinx + 3 sin2x + sinx + 1 50 Find the range of f(x) =  Watch Free Video Solution on Doubtnut

  12. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Draw the graph of y = (sin2x)√1 + tan2x , find its domain and range. 51  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the domain of f(x) = sin-1x2 52 . 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of f(x) = sin- 1x + tan-1x + cos-1x. 53  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain of f(x) =√cos- 1x - sin-1x 54  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 55

  13. ( ) 2x Find the range of tan-1 1 + x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the domain for f(x) = sin- 11 + x2 56 2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the range of f(x) = cos-1√1 + 2x2 57 1 + x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions sin-1x x 58 Find the domain of the function: f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain of the function: f(x) = sin-1(|x - 1| - 2) 59  Watch Free Video Solution on Doubtnut

  14. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the domain of the function: f(x) = cos-11 + 3x + 2x2 60  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions sin-1(x - 3) √9 - x2 Find the domain of the function: f(x) = 61  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) ( ) Find the domain of the function: f(x) = cos-16 - 3x + cosec-1x - 1 4 2 6 − 3x f(x) = cos−1( ) 62 4 x − 1 + cosec−1( ) 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions sec-12 - |x| √ ( ) 63 Find the domain of the function: f(x) = 4  Watch Free Video Solution on Doubtnut

  15. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the range of f(x) = tan- 1√ x2- 2x + 2 64  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions √ cos-1√ 1 - x2- sin-1x ( ) 65 Find the range of f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the range of the function f(x) = cot- 1(log)0.5x4- 2x2+ 3 66  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions √( ) 1 - 5x 7-x- 7 67 Find the domain of f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions

  16. 68 Find the domain of f(x) =√(0.625)4-3x- (1.6)x(x+8)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions tan- 1x ( ) -1 69 Find the domain of f(x) = (log)10(log)2(log)2 π  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain and range of f(x) = (log)3{cos(sinx)} √ 70  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve logxx2- 1 ≤ 0 ( ) 71  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the number of solution of 2x+ 3x+ 4x- 5x= 0 72  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions

  17. { ( )} x2 4 73 Find the domain of f(x) = sin-1(log)9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions [ { ( }] Find the domain of function f(x) = (log)4(log)5(log)318x - x2- 77 f(x) 74 = (log)4[(log)5{(log)3(18x − x2− 77}]  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 0,π ( ) 1 . Then find the domain of the function f(x) = Let x ∈ 75 2 √( ) - (log)sinxtanx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions (log)0.4 √ ( ) x - 1 x + 5 76 Find the domain of f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) | (log)ex2 | 77 Find the range of f(x) = (log)ex - (log)ex  Watch Free Video Solution on Doubtnut

  18. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 4x+ 8 √ ( ) 2 78 (x-2)- 13 - 22(x-1) Find the domain of the function : f(x) = 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the domain of the function : f(x) = sin-1(log)2x 79  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the domain of the function : f(x) = (log)( x- 4)x2- 11x + 24 80  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 3 ( ) 4 - x2+ (log)10x3- x 81 Find the domain of the function : f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions √ (log)0.3|x - 2| |x| Find the domain of the function : f(x) = 82

  19.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions √ {( )} (log)10x 83 Find the domain of the function : f(x) = (log)10 2 3 - (log)10x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 Find the domain of the function : f(x) = x2- 7x + 13 ( ) 84 (log)1 √ 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) sinx - cosx + 3√2 √2 85 Find the range of f(x) = (log)2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions

  20. (log)24sin2x - 2√3sinx - 2sinx +√3 + 1 ( ) √ Find the value of x in [ - π, π] for which f(x) = 86 is defined.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain of f(x) =√([x] - 1) +√(4 - [x]) (where [ ] represents the greatest integer function). 87  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain and range of f(x) = sin- 1[x]wher[] represents the greatest function). 88  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the domain and range of f(f) = log{x}, where{} represents the fractional part function). 89  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of f(x) = [sin{x}], where {} represents the fractional part function and [] represents the greatest integer function. 90  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types

  21. Of Functions 91 Solve 2[x] = x + {x},whre[]and{} denote the greatest integer function and the fractional part function, respectively.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions x - [x] 1 - [x] + x′, where[] represents the greatest integer function. 92 Find the range of f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve x2- 4 - [x] = 0 (where [] denotes the greatest integer function). 93  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions If f(x) = [x],0 ≤ {x} < 0.5 and f(x) = [x] + 1,0.5 < {x} < 1 then prove that f (x) = -f(-x) (where[.] and{.} represent the greatest integer function and the fractional part function, respectively). 94  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions In the questions, [x]and{x} represent the greatest integer function and the fractional part function, respectively. Solve: [x]2- 5[x] + 6 = 0. 95  Watch Free Video Solution on Doubtnut

  22. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions In the questions, [x]and{x} represent the greatest integer function and the fractional part function, respectively. If y = 3[x] + 1 = 4[x - 1] - 10, then find the value of [x + 2y]. 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 1 Find the domain of f(x) = √x - [x] (b) f(x) = log[x]f(x) = log{x} 97  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 Find the domain of f(x) = √|[|x| - 1]| - 5 98  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 1 Find the domain of 99 √|[|x| - 1]| - 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions √(1 - sinx) ( +cos-1(1 - {x}). Find the domain of f(x) 100 ) (log)51 - 4x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) 101 Find the range of f(x) = cos (log)e{x} .  Watch Free Video Solution on Doubtnut

  23. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions (log)[x] √ ( ) |x| x 102 Find the domain and range of f(x) = cos-1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Find the range of f(x) = (log)[x-1]sinx. 103  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Solve (x - 2)[x] = {x} - 1, (where [x]and{x} denote the greatest integer function less than or equal to x and the fractional part function, respectively). 104  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 105 Write the equivalent (piecewise) definition of f(x) = sgn(sinx).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions ( ) Find the range of f(x) = sgn x2- 2x + 3. 106  Watch Free Video Solution on Doubtnut

  24. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions 107 Verify that xsgnx = |x|; |x|sgnx = x; x(sgnx)(sgnx) = x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions If 2 min .{f(x) - g(x),0} = f(x) - g(x) - |g(x) - f(x)| f:R → R and g: R → R are two given functions, then prove that 108  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions . sinx,cos2x,x ∈ [0, 2π]. Write the equivalent Draw the graph of the function f(x) = max definition of f(x) and find the range of the function. 109  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions . Consider the function: f(x) = max of f(3). 1, |x - 1|, min {4, |3x - 1|}}∀x ∈ R . Then find the value 110  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions x2,(1 - x)2, 2x(1 - x) where 0 ≤ x ≤ 1 { } 111 Find the equivalent definition of f(x) = max

  25.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Write the equivalent definition and draw the graphs of the following functions. ( ) 112 f(x) = sgn (log)e|x|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Functions Write the equivalent definition and draw the graphs of the following functions. ( ) 113 f(x) = sgn x3- x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) x2+ 4x + 7 x2+ x + 1 → Rwheref(x) = 114 Let f:R, . Is f(x)oneone?  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) If f:X → [1, ∞) is a function defined as f(x) = 1 + 3x3, find the superset of all the sets X

  26. 115 such that f(x) is one-one.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) 116 Let f:R → R, where f(x) = sinx, Show that f is into.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) → Z be a function defined as f(x) = x - 1000. Show that f is an into function. 117 Let f:N  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) → R defined by f(x) = (x - 1)(x - 2)(x - 3) is surjective but not injective. 118 Show that f:R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) x2 →A given by f(x) = is surjection, then find A. If the function f:R 119 x2+ 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions)

  27. 120 If f:R→ is a function such that f(x) = x3+ x2+ 3x + sinx, then identify the type of function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) x2- 4 x2+ 1 →R is given by f(x) = 121 If f:R , identify the type of function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) Which of the following function from Z to itself are bijections? f(x) = x3 (b) f(x) = x + 2 f(x) = 2x + 1 (d) f(x) = x2+ x 122  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) { n - 1 2 n 2; when n is even Identify the type of If f:N → Zf(n) = ; when n is odd = - 123 function  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) x2- 4 x2+ 1 →R is given by f(x) = If f:R , identify the type of function. 124  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) →S, defined by f(x) = sinx -√3cosx + 1,ison → , then find the set S. 125 If f:R  Watch Free Video Solution on Doubtnut

  28. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different TypesCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) [ ] 2x Let f:( - 1, 1) → B be a function defined by f(x) = tan-1 1 - x2 . Then f is both one- ) [ 126 [ ) ( ) ( ] one and onto when B is the interval. (a) 0,π (b) 0,π π 2,π π 2,π (c) - (d) - 2 2 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Different Types Of Mappings (Functions) ( ) 0,π ( ) x2- k 1 + x2 . Then find the possible values be defined by g(x) = cos- 1 Let g: R → 127 3 of k for which g is a surjective function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions Which of the following function is (are) even, odd, or neither? f(x) = x2sinx ( ) f(x) = log x +√1 + x2 ( ) 1 - x 1 + x f(x) =√1 + x + x2-√1 - x + x2 f(x) = log 128 ex+ e-x 2 f(x) = sinx - cosxf(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions f(x) = h1(x) - h1( - x) ( h2(x) - h2( - x) ... h2n+1(x) - h2n+1( - x)andf(200) = 0, )( ) ( If then 129

  29. prove that f(x) is many one function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions ( )( ) Check whether the function h(x) = √sinx -√tanx √sinx +√tanx is whether odd or even. 130  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions Find whether the given function is even or odd: f(x) = xsinx + tanx ; where [] denotes x+ π π [ ] 1 2 131 - the greatest integer function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions 132 if`f(x)={x^3+x^2,for0lt=xlt=2x+2,for2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions Let the function f(x) = x2+ x + s ∈ x - cosx + log(1 + |x|) be defined on the interval [0,1] .Define functions g(x)andh(x) ∈ [ - 1,0] g( - x) = - f(x)andh( - x) = f(x)∀x ∈ [0, 1]. 133 satisfying  Watch Free Video Solution on Doubtnut

  30. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions 134 video and ocr didnt match  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions ( ) x4+ x2+ 1 x2+ x + 1 135 Identify the following functions whether odd or even or neither: f(x) = log  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions Identify f(x) = xg(x)g( - x) + tan(sinx) the following functions whether odd or even or neither: 136  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions [| |] sinx 2 Identify the following functions whether odd or even or neither: f(x) = cos|x| + 137 where [.] denotes the greatest integer function.  Watch Free Video Solution on Doubtnut

  31. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions ( ) Identify the following functions whether odd or even or neither: f(x) = log x +√x2+ 1 138  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Even And Odd Functions Identify f(x) = {x|x|f or x ≤ - 1; [1 + x] + [1 - x]f or xε( - 1,1); - x|x|f or x ≥ 1 the following functions whether odd or even or neither: 139  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Find f(x) = elog(sinx)+ tan3x - cosec(3x - 5) the period (if periodic) of the following function 140  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Find the period (if periodic) of the following function ([.] denotes the greatest integer functions): f(x) = x - [x - b],b ∈ R 141  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Find the period (if periodic) of the following function ([.] denotes the greatest integer |sinx + cosx| |sinx| + |cosx| 142 functions): f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Find the period (if periodic) of the following function ([.] denotes the greatest integer tanπ 2 143 functions): f(x) = [x]  Watch Free Video Solution on Doubtnut

  32. CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic 144 Find the period f(x) = sinx + {x}, where {x} is the fractional part of x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic 145 If f(x) = sinx + cosax is a periodic function, show that a is a rational number  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Discuss whether the function f(x) = sin(cosx + x) is period or not. If yes, then what is its period? 146  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic 147 Find the period of cos(cosx) + cos(sinx).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic ( ) 5x n For what integral value of n is 3π the period of the function cos(nx)sin ? 148  Watch Free Video Solution on Doubtnut

  33. CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Let [x] denotes the greatest integer less than or equal to x . If the function f(x) = tan√[n]x has period π 3 149 ( ) . then find the value of n.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic sin(πx) n! cos(πx) (n + 1)! 150 Find the period of f(x) = -  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic Find the period of f(x) = sinx +tanx +sinx 22+ tanx sinx 2n-1+ tanx 2n 151 23+ + 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Functions AND FUNCTIONS_Periodic If f(x) = λ|sinx| + λ2|cosx| + g(λ) has a period = π 152 2 then find the value of λ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite Find the fundamental period of f(x) = cosxcos2xcos3x. 153  Watch Free Video Solution on Doubtnut

  34. CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite Let f:{2,3, 4,5} → {3, 4,5, 9}a n dg:{3,4, 5,9} → {7, 11, 15} be functions defined at f(2)=3,f(3)=4,f(4)=f(5)=5,g(3)=g(4)=7,a n dg(5)=g(9)=11. Find g(f(x)) 154  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite → → 155 Let f(x)andg(x) be bijective functions where f:{1,b, c, d} respectively. Then, find the number of elements in the range of g(f(x)}. 1, 2,3, 4andg:{3,4, 5,6} 2,x,y, z,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite Let f(x) = ax + bandg(x) = cx + d, a ≠ 0. Assume a = 1, b = 2. If (fog)(x) = (gof)(x) for all x, what can you say about candd? 156  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite Suppose that g(x) = 1 +√x and f(g(x)) = 3 + 2√x + x. Then find the function f(x). 157  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite 158 The function f(x) is defined in [0, 1] . Find the domain of f(tanx).

  35.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite If f is the greatest integer function and g is the modulus function, then find the value of ( ) ( ) 159 5 3 5 3 . (g0f) - - (fog) -  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite Let f(x) = {1 + |x|, x < - 1[x], x ≥ - 1 , where [.] denotes the greatest integer function. The find the value of f{f( - 2, 3)}. 160  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite [ ] [ ] 1 + x 1 - x 2x 161 = 2f(x). If f(x) = log , then prove that f 1 + x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite If the domain of y = f(x)is[ - 3, 2], then find the domain of g(x) = f(|[x]|), wher[] denotes the greatest integer function. 162  Watch Free Video Solution on Doubtnut

  36. CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite ( ) Let f be a function defined on [0,2]. Then find the domain of function g(x) = f 9x2- 1 163  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite 164 if `f(x)={ lnx, 0=1`  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite ( ) π 4 Let f(x) = tanxandg(f(x)) = f x - , where f(x)andg(x) are real valued functions. Prove 165 ( ) x + 1 x + 1 . that f(g(x)) = tan  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Composite A function f has domain [ - 1,2] and rang [0,1] . Find the domain and range of the function g defined by g(x) = 1 - f(x + 1). 166  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse Which of the following functions has inverse function? f:Z → Zdefined byf(x) = x + 2 f:Z → Zdefined byf(x) = 2xf:Z → Zdefined byyf(x) = xf:Z → Zdefined byf(x) = |x| 167  Watch Free Video Solution on Doubtnut

  37. CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse x - 2 x - 3 is f invertible? → B be defined by f(x) = Let A = R - {3}, B = R - {1}, and let f:A 168 Explain.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse Let f:R → R be defined by f(x) = ex- e- x/2. Is f(x) invertible? If so, find is inverse. ( ) 169  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse ( ) 3, then find the function g such that f(g(x)) = g(f(x)). If f(x) = ax2+ b 170  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse If f(x) = 3x - 2and(gof)-1(x) = x - 2, then find the function g(x). 171  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse { x, < 1x2, 1 ≤ x ≤ 48√x, x > 4 172 Find the inverse of f(x) =

  38.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse 1 2+ 3 4 x -3 √ Solve the equation x2- x + 1 = . 4, wherex ≥ 173  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse { - x,x < 0x2, 1 ≤ x ≤ 48√xx > 4 174 Find the inverse of the following f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse 1 2+ 3 4 3 4,wherex ≥ √ Solve the equation: x2- x + 1 = x - 175  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse Find the inverse of the function: f:R → ( - ∞,1)givenbyf(x) = 1 - 2-x 176  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse

  39. → 0,1def ∈ edbyf(x) = x - [x],where[.] represents the greatest integer function. 177 f:(2,3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse x3- 1, ,x < 2x2+ 3, x ≥ 2 { Find the inverse of the function: f(x) = 178  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse 179 Find the inverse of the function: f:[ - 1, 1] → [ - 1,1]defined byf(x) = x|x|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Inverse [ 1 2,∞ , wheref(x) = 2x( x-2) ] 180 Find the inverse of the function: f:( - ∞, 1] →  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Identical 1 181 Find the value of x for which function are identical. f(x) = xandg(x) = 1/x  Watch Free Video Solution on Doubtnut

  40. CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Identical 1 Find the value of x for which function are identical. f(x) = cosxandg(x) = 182 √1 + tan2x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Identical Find the value of x for which function are identical. f(x) =√9 - x2 9 - x2 x - 2 √ 183 andg(x) = √x - 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Function AND FUNCTIONS_Identical Find f(x) = tan-1x +tan-11 the value of x for which function are identical. 184 andg(x) = sin-1x + cos-1x x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 185 Plot y = |x|, y = |x - 2|, and y = |x + 2|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 186 Plot y = sinxandy = sin2x.

  41.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs ( ) x 2 Plot y = sinxandy = sin 187  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 2x + 3,x ≤ 1 - x2+ 6,x > 1 { 188 Consider the function f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 189 Plot y = sinxandy = 2sinx.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs Plot the curve y = (log)e( - x). 190  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 191 Draw the graph for y = |logx|  Watch Free Video Solution on Doubtnut

  42. CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_TransformationCENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 192 Sketch the graph for y = |sinx|.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs Sketch the curve y = log|x| 193  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 194 Sketch the curve |y| = (x - 1)(x - 2).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 195 Draw the graph of the function: f(x) = log|x|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 196 Draw the graph of the function: f(x) = ||x - 2| - 3|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs

  43. 197 Draw the graph of the function: |f(x)| = tanx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs | | Draw the graph of the function: f(x) = x2- 3 x| + 2| 198  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 1 199 Draw the graph of the function: f(x) = - |x - 1| 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs 200 Find the total number of solutions of sinπx = |ln|x ∣ ∣  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs | | x2 x - 1 201 Draw the graph of the function: Solve ≤ 1 using the graphical method.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation

  44. Of Graphs 202 Given the graph of y = f(x), which of the following is graph of y = f(1 - x)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS AND FUNCTIONS_Transformation Of Graphs Which of the following cosx secx+ pair(s) sinx cosecx of function have same graphs? secx cosx- tanx cotx,g(x) = 203 f(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional . Iff(0) = 1, f ∈ df(100). 204 Let f(x) = x + f(x - 1)f or ∀x ∈ R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional Consider 2f(xy) = (f(x))y+ (f(y))x∀x,y ∈ Rand(1) = a, wherea ≠ 1. a real-valued function f(x) satisfying (a - 1) Prove that 205

  45. n ∑ i=1 f(i) = an+ 1- a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional Let a function f(x) satisfies f(x) + f(2x) + f(2 - x) + f(1 + x) = x, ∀x ∈ R. Then find the value of f(0). 206  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional f(x) y Let f be a function satisfying of x. Then f(xy) = for all positive real numbers xandy. If 207 f(30) = 20, then find the value of f(40).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional . ( ) ( ) 1 x 1 x 208 If f(x) is a polynomial function satisfying and f(x) f = f(x) + f f(4) = 65,thenf ∈ df(6).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional ( ) 2002 x = 3x. Then find f(x). Let f be a real-valued function such that f(x) + 2f 209  Watch Free Video Solution on Doubtnut

  46. CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional ( ) 1 x →R is an odd function such that f(1 + x) = 1 + f(x)x2f 210 = f(x),x ≠ 0 then find f(x). If f:R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional ( y+ 1 for x, y > 0. ) . 1 x+1 Let f:R+→ R be a function which satisfies f(x) f(y) = f(xy) + 2 211 Then find f(x).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional f(x)onR→ A continuous function satisfies the relation 212 . f(x) + f(2x + y) + 5xy = f(3x - y) + 2x2+ 1f or ∀x,y ∈ R Thenf ∈ df(x).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional 213 Prove that f(x)givenbyf(x + y) = f(x) + f(y)∀x ∈ R is an odd function.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional . If f(x + y) = f(x) f(y) for all real x, yandf(0) ≠ 0, then prove that the function

  47. 214 f(x) 1 + {f(x)}2 is an even function. g(x) =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional Let f(x) f(x + k) + f(x) = 0f or allx ∈ R. Prove that f(x) is periodic with period 2k. be periodic and k be a positive real number such that 215  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional If f(x) satisfies the relation f(x) + f(x + 4) = f(x + 2) + f(x + 6) for allx, then prove that f(x) is periodic and find its period. 216  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional An odd function is symmetric about the vertical line x = a, (a > 0),and if ∞ ∑ r=0 217 f(1 + 4r]r= 8, then find the value of f(1). [  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional Check whether the function defined by f(x + λ) = 1 +√2f(x) - f2(x)∀x ∈ R is periodic or not. If yes, then find its period (λ > 0). 218

  48.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional If for all real values of uandv, 2f(u)cosv = (u + v) + f(u - v), prove that for all real values of x, f(x) + f( - x) = 2acosx. f(π - x) + f( - x) = 0 f(π - x) + f(x) = 2bsinx. Deduce that f(x) = acosx + bsinx,wherea, b are arbitrary constants. 219  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional 9x Let ( . Show ) f(x) + f(1 - x) = 1 ) and, hence, evaluate. f(x) = ) ( 9x+ 3 220 ) ( ( 1 2 3 1995 1996 f + f + f + + f 1996 1996 1996  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional { } 2 and f(0) = 1∀x, y ∈ R, determ ∈ ef(n),n ∈ N. 221 If f(x + y + 1) = √f(x) +√f(y)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional . g(b)∀a,b ∈ R. If zero is not an element in Let g(x) be a function such that g(a + b) = g(a) 222 . g( - x). the range of g, then find the value of g(x)  Watch Free Video Solution on Doubtnut

  49. CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional 223 If f(x + 2a) = f(x - 2a),thenprovethatf(x)isperiodi ⋅  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional 224 If f(x + f(y)) = f(x) + y∀x, y ∈ Randf(0) = 1, then find the value of f(7).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional ( ) 1 x If f:R+→ 225 = 2(x + 1),thenf ∈ df(x). R, f(x) + 3xf  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional f:R → R, f x2+ x + 3 + 2f x2- 3x + 5 = 6x2- 10x + 17∀x ∈ R, then find the function ( ) ( ) 226 f(x).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional f:R+→ Rsuch ( )( ) Consider thatf(3) = 1 for . 227 3 x 3 y a ∈ R+andf(x) = 2f(xy)∀x,y ∈ R+. Then find f(x). f(y) + f f  Watch Free Video Solution on Doubtnut

  50. CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional 228 Determine all functions f:R → R such that f(x-f(y))=f(f(y))+xf(y)+f(x)-1 ∀x, y ≥ 0 ∈ R.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional Determine the function satisfying f2(x + y) = f2(x) + f2(y)∀x, y ∈ R. 229  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional →R is a function satisfying the property f(2x + 3) + f(2x + 7) = 2∀x ∈ R, then find the If f:R fundamental period of f(x). 230  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional ( ) . 1 x If f(x) is an even function and satisfies the relation x2 = g(x), whereg(x) is an f(x) - 2f 231 odd function, then find the value of f(5).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_CALCULUS_RELATIONS Equation AND FUNCTIONS_Functional A f(x - y) = f(x)f(y) - f(a - x)f(a + y), where a is a given constant and f(0), f(2a-x) = real valued function f(x) satisfies the functional equation 232  Watch Free Video Solution on Doubtnut

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