Math 1 - Calculus - Sheet (1) PDF Fall 2024

Faculty of Engineering Department of Engineering Mathematics and physics...

Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 Math 1 - Calculus - Sheet (1) Functions – part A This sheet covers the following topics: - Power function - Polynomials - Rational functions - Absolute function - Even & odd functions - Composite functions - Transformation of functions - Trigonometric functions Essay problems Find the domain of the following functions: Ans. 1) 𝑦 = √2𝑥 + 10 [−5, ∞[ 𝑥+1 2) 𝑦 = ] − ∞, 3[ √3−𝑥 √𝟑𝒙−𝟗 3) 𝑦 = [3, ∞[−{4} 𝒙𝟐 −𝟒𝒙 3 4) 𝑟 = √𝜃 − 1 𝑅 √1−𝑡 5) 𝑧 = ] − ∞, 1] − {−2} 𝑡 2 −2𝑡−8 6) 𝑦 = √𝑥 2 + 2𝑥 − 3 ] − ∞, −3] ∪ [1, ∞[ √𝑥 2 −𝑥−6 7) 𝑦 = ] − ∞, −2] ∪ [3,5[ ∪ ]5, ∞[ 𝑥−5 𝑥+2 8) 𝑦 = √ [−2,3[ 3−𝑥 𝑥+2 9) 𝑦 = √ ] − ∞, −4[ ∪ [−2, ∞[ 𝑥+4 √𝑥+2 10) 𝑦 = [−2, ∞[ √𝑥+4 Solve the following inequalities: Ans. 11) |𝑥 + 5| ≥ 1 ] − ∞, −6] ∪ [−4, ∞[ 12) |3 − 𝑥| < 2 ]1,5[ 13) |𝑥 2 + 2𝑥 − 1| < 2 ] − 3,1[ −{−1} 1  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 14) |𝑥 2 − 4𝑥 − 4| ≥ 8 ] − ∞, −2] ∪ [6, ∞[ ∪ {2} Find the domain of the following functions: 15) 𝑦 = √3 − |𝑥 − 2| [−1,5] 1 16) 𝑦 = 𝑅 √5+|𝑥−1| 𝜃+1 17) 𝑟 = 𝑅 − {3, −3} √𝜃2 −3 √√𝑡 2 −2 18) 𝑠 = ] − ∞, −2[ ∪ [2,3[ ∪ ]3, ∞[ 𝑡 2 −𝑡−6 19) Let 𝑓(𝑥) be some given function and define 1 𝑔(𝑥) = [𝑓(𝑥) + 𝑓(−𝑥)] 2 1 ℎ(𝑥) = [𝑓(𝑥) − 𝑓(−𝑥)] 2 Show that 𝑔(𝑥) is an even function and ℎ(𝑥) is an odd function. Hence, deduce that any function 𝑓(𝑥) can be written as a sum of an even function and an odd function. 20) Show that if 𝑓(𝑥) and 𝑔(𝑥) are two even functions and ℎ(𝑥) is an odd function, then (𝑓 𝑜 𝑔 𝑜 ℎ)(𝑥) is an even function. In each of the following problems, starting from 𝑓(𝑥), give a sequence of transformations to produce 𝑔(𝑥), and hence sketch 𝑔(𝑥): 21) 𝑓(𝑥) = √𝑥 , 𝑔 ( 𝑥 ) = 2 √𝑥 + 1. 22) 𝑓(𝑥) = √𝑥, 𝑔(𝑥) = √2 − 4𝑥 23) 𝑓(𝑥) = |𝑥|, 𝑔(𝑥) = 1 − |2𝑥 + 1|. 1 2 24) 𝑓(𝑥) = 𝑥, 𝑔(𝑥) = | 𝑥−4 | 25) 𝑓(𝑥) = 𝑥 3 , 𝑔(𝑥) = |(1 − 𝑥)3 | 26) 𝑓(𝑥) = 𝑥 2 , 𝑔(𝑥) = (4 − 2𝑥)2. 27) 𝑓(𝑥) = 𝑥 2 𝑔(𝑥) = 𝑥 2 + 2𝑥 − 3. (Hint: 𝑥 2 + 2𝑥 − 3 = (𝑥 + 1)2 − 4) 28) 𝑓(𝑥) = sin 𝑥 𝑔(𝑥) = |3 sin 2𝑥 | 2  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 𝜋 29) 𝑓(𝑥) = cos 𝑥, 𝑔(𝑥) = cos(2𝑥 + 2 ) Sketch the graph of each of the following function over a complete period: 30) 𝑓(𝑥) = 3 sin 𝜋𝑥 31) 𝑓(𝑥) = 2 sin2 2𝑥 𝜋 32) 𝑓(𝑥) = 3 cos (2𝑥 − 2 ) Find amplitude, period and frequency of the following functions: 33) 𝑦 = −2 cos(𝜋 − 3𝑥). 34) 𝑦 = sin 2𝑥 − 2 cos 2𝑥. 35) 𝑟 = 6 cos2 2𝜋𝑥. Find the fundamental period of the following functions: 𝜋𝑥 36) 𝑦 = 2 sin 𝜋𝑥 + 3 cos 3 37) 𝑦 = sin 𝑥 cos 𝑥 + cos 4𝑥. 38) 𝑦 = tan 2𝜋𝑥 − sin 2𝜋𝑥 3  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 Answers 21) √𝑥 → √𝑥 + 1 → 2√𝑥 + 1 Horizontal shift -1 Vertical expansion 2 22) √𝑥 → √−𝑥 → √−(𝑥 − 1/2) → 2√−(𝑥 − 1/2) = √2 − 4𝑥 Reflection about y-axis Horizontal shift 2 Horizontal compression 4 23) 1 1 |𝑥| → |𝑥 + | → 2 |𝑥 + | = |2𝑥 + 1 | 2 2 → −|2𝑥 + 1| → 1 − |2𝑥 + 1| Horizontal shift −1 Horizontal compression 2 Reflection about x-axis Vertical shift +1 24) 1 1 1 1 →| |→| | → 2| | 𝑥 𝑥 𝑥−4 𝑥−4 2 =| | 𝑥−4 Absolute 𝑦 Horizontal shift +4 Vertical expansion 2 4  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 25) 𝑥 3 → (𝑥 − 1)3 → |(𝑥 − 1)3 | = |(1 − 𝑥)3 | Horizontal shift +1 Absolute 𝑦 26) 𝑥 2 → (𝑥 − 4)2 → (2𝑥 − 4)2 = (4 − 2𝑥)2 Horizontal shift +4 Horizontal compression +2 Or 𝑥 2 → (𝑥 − 2)2 → 4(𝑥 − 2)2 Horizontal shift +2 Vertical expansion +4 27) 𝑥 2 → (𝑥 + 1)2 → (𝑥 + 1)2 − 4 Horizontal shift -1 Vertical shift -4 5  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 28) sin 𝑥 → sin 2𝑥 → 3 sin 2𝑥 → |3 sin 2𝑥| Horizontal compression 2 Vertical expansion 3 Absolute 𝑦 29) 𝜋 cos 𝑥 → cos 2𝑥 → cos 2(𝑥 + ) 4 𝜋 = cos (2𝑥 + ) 2 Horizontal compression 2 −𝜋 Horizontal shift 4 (30) (31) (32) (33) 2π 3 |A| = 2, T = ,𝑓 = 3 2𝜋 (34) 1 |A| = √5, T = π, 𝑓 = 𝜋 (35) 1 |A| = 3, T = , 𝑓 = 2 2 (36) 6 (37) 𝜋 (38) 1 6  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 MCQ problems 1 *) For the function f(x) = 𝑥−1 : √ 𝑥−2 (1) The domain can be determined by the following condition: 𝑥−1 (a) (𝑥 − 1) ≥ 0 & (𝑥 − 2) > 0 (b) (𝑥 − 1) > 0 & (𝑥 − 2) > 0 (c) 𝑥 − 2 > 0 𝑥−2 𝑥−2 (d) 𝑥 − 1 ≥ 0 (e) 𝑥 − 1 > 0 (f) (𝑥 − 2) ≠ 0 (2) The domain is: (a) [1,2[ (b) ]1,∞ [ (c) ]1,2[ (d) 𝑅 - ]1,2] (e) ]2,∞ [ (f) 𝑅 - [1,2] -------------------------------------------------------------------------------------------------------------------------------------- (𝑥+1)(𝑥−3) (3) The domain of is: 𝑥√(𝑥+1)(2−𝑥) (a) ] - ∞ , -1 [ ∪ ]2 , ∞ [ (b) ] -1, 2[ - {0} (c) ]-1 , 2[ (d) ([-1 , 2[ ∪ {3}) - {0} -------------------------------------------------------------------------------------------------------------------------------------- (4) For the function f(x) = √−3 − √𝑥 2 the domain is: (a) ∅ (b) ] -∞ , -3] (c) 𝑅 - {-3} (d) 𝑅 -------------------------------------------------------------------------------------------------------------------------------------- (5) The domain of 𝑓(𝑥) = √3 + √(𝑥)2 is: (a) 𝑅 (b) ∅ (c) ] − ∞ , −3] (d) 𝑅 − {−3} -------------------------------------------------------------------------------------------------------------------------------------- 𝑥−1 (6) The domain of √(𝑥+4)(𝑥+1) is: (a) (𝑅 - [-1,1[) - {-4} (b) 𝑅 - {-1, -4} (c) [1, ∞ [ (d) 𝑅 - [-1, 1[ (e) ]1, ∞ [ (f) ]-4, -1[ ∪ [1, ∞ [ -------------------------------------------------------------------------------------------------------------------------------------- √𝑥 2 +4𝑥+3 (7) The domain of f(x) = is: √𝑥+2 (a) [-1, ∞ [ (b) ]-2, -1[ c) [-2, ∞ [ (d) [-3, -2[ ∪ [-1, ∞ [ -------------------------------------------------------------------------------------------------------------------------------------- 7  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 1 (8) The domain of the function f(x) = is: √𝑥 2 (𝑥−1) (a) ]1, ∞ [ (b) 𝑅 − {0,1} (c) 𝑅 (d) ]-∞ ,0[ ∪ ]1, ∞ [ (e) none of the above *) For 𝑓(𝑥) = √4 − |4 − 𝑥| (9) The domain 𝐷 of 𝑓(𝑥) can be obtained by solving: (a) 4 − |4 − 𝑥| > 0 (b) √4 − √|4 − 𝑥| ≥ 0 (c) 4 − |4 − 𝑥| > 1 (d) 4 − |4 − 𝑥| ≥ 1 (e) 4 − |4 − 𝑥| ≥ 0 (f) |4 − 𝑥| ≥ 0 (10) Hence, the domain D of 𝑓(𝑥) is: (a) 𝑅 − {4} (b) [1, 7] (c) [0, 8] (d) ]1, 7[ (e) [4, ∞ [ (f) 𝑅 + 1 (11) the domain of 𝑓(𝑥) is: (a) D - [1, 7] (b) ]1,7[ (c) D - ]1,7[ (d) 𝑅 - {3,5} (e) D - {1,7} (f) ]0, 8[ -------------------------------------------------------------------------------------------------------------------------- 1 (12) For the function f(x) = the domain is: √√𝑥+ |𝑥−1| (a) 𝑅 + ∪ {0} (b) [1, ∞ [ (c) 𝑅 (d) 𝑅 − {0} -------------------------------------------------------------------------------------------------------------------------- (13) The domain of 𝑓(𝑥) = √3 + √(𝑥 + 1)2 is: (a) 𝑅 (b) [2 , ∞[ (c) ] − 3 , −1[ ∪ [2 , ∞[ (d) 𝑅− ]2 , ∞[ -------------------------------------------------------------------------------------------------------------------------- (14) The domain of 𝑓(𝑥) = √3 − √(𝑥 + 1)2 is: (a) 𝑅 (b) [−4 ,2] (c) ]2 , ∞[ (d) 𝑅− ] − 4 , 2[ -------------------------------------------------------------------------------------------------------------------------- (15) The domain of 𝑓(𝑥) = √4 − √𝑥 − 2 is: (a) 𝑅 (b) [2,18] (c) ] − ∞ , 2] ∪ [18, ∞[ (d) 𝑅 − {2} -------------------------------------------------------------------------------------------------------------------------- (16) The domain of 𝑓(𝑥) = √4 + √𝑥 − 2 is: (a) 𝑅 (b) [2,18] (c) ] − ∞ , 2] ∪ [18, ∞[ (d) [2 , ∞[ -------------------------------------------------------------------------------------------------------------------------- 8  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 1 (17) The domain of 𝑓(𝑥) = √1 + 𝑥 is: (a) 𝑅 − {0} (b) 𝑅− ] − 1,0] (c) [−1,0] (d) [−1, ∞[ (e) ] − 1, ∞[ -------------------------------------------------------------------------------------------------------------------------- (18) Which of the following relations does not represent a function? (a) 𝑦 = 𝑥 2 (b) 𝑥 = sin 𝑦 (c) 𝑦 = cos 𝑥 (d) 𝑦 = 𝑥 3 (19) If 𝑓(𝑥) is an odd function, and 𝑔(𝑥) is an even function, then 𝑔(𝑓(𝑥)) is: (a) can’t be determined (b) even (c) odd (d) neither even nor odd -------------------------------------------------------------------------------------------------------------------------- (20) If 𝑓(𝑥) = 3𝑥 − 6 and 𝑔(𝑥) = 𝑥 3 + 2 then the function f(g(x)) is: (a) can’t be determined (b) even (c) odd (d) general -------------------------------------------------------------------------------------------------------------------------- 3 3 *) If 𝑓(𝑥) = √𝑥 2 + 1 , 𝑔(𝑥) = √𝑥 3 + 𝑥 and ℎ(𝑥) = √𝑥 3 + 2𝑥 then: (21) (𝑓 𝑜 𝑔 ) (x) is: (a) can’t be determined (b) even (c) odd (d) general (22) (𝑓 𝑜 ℎ ) (x) is: (a) can’t be determined (b) even (c) odd (d) general -------------------------------------------------------------------------------------------------------------------------- (23) If 𝑓(𝑥) is general function, then 𝑔(𝑥) = 𝑓(𝑥)𝑓(−𝑥) is: (a) can’t be determined (b) even (c) odd (d) general -------------------------------------------------------------------------------------------------------------------------- (24) For a function 𝑓 where 𝑓(𝑥) ≠ 𝑐, then the function 𝑔(𝑥) = 𝑓(𝑎 + 𝑥) − 𝑓(𝑎 − 𝑥) is: (a) can’t be determined (b) even (c) odd (d) general -------------------------------------------------------------------------------------------------------------------------- (25) The function 𝑓(𝑥) = (1 + 𝑥 2 )5 is: (a) odd (b) one-to-one (c) even (d) none of the above -------------------------------------------------------------------------------------------------------------------------- 𝑥−3 (26) If 𝑓(𝑥) = , 𝑔(𝑥) = 𝑥 2 − 4 then (𝑓 𝑜 𝑔)(𝑥) is: 2𝑥 9  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 𝑥−3 2 𝑥 2 −7 (𝑥−3)(𝑥2 −4) (𝑥−3) (a) ( 2𝑥 ) − 4 (b) 2(𝑥 2−4) (c) (d) 2𝑥(𝑥 2−4) 2𝑥 (27) If 𝑓(𝑥) is an even function, and 𝑓(𝑔(𝑥)) is an even function, then 𝑔(𝑥) may be (a) odd (b) even (c) general (d) even or odd (e) even or general (f) even or odd or general ---------------------------------------------------------------------------------------------------------------- (28) Given the graph of 𝑓(𝑥) then the graph of 𝑔(𝑥) = 𝑓(𝑎𝑥), 𝑎 > 1 can be obtained from 𝑓(𝑥) by (a) Vertical shift (b) Horizontal expansion (c) Vertical expansion (d) Horizontal shift (e) Horizontal compression (f) Vertical compression -------------------------------------------------------------------------------------------------------------------------- (29) Given the graph of the function 𝑓(𝑥) = 3 − √4 + 𝑥 , then the graph of 𝑔(𝑥) = 3 − √4 − 𝑥 can be obtained by (a) Reflection about x-axis (b) Reflection about y-axis (c) Vertical shift +3 (d) Horizontal shift -4 -------------------------------------------------------------------------------------------------------------------------- (30) Given the graph of the function 𝑓(𝑥) = 𝑥 2 − 6𝑥 , then the graph of 𝑔(𝑥) = (𝑥 − 3)2 + 1 can be obtained by (a) Vertical shift +1 (b) Horizontal shift +3 (c) Reflection about x-axis (d) Vertical shift +10 -------------------------------------------------------------------------------------------------------------------------- *) Given the graph of the function 𝑓(𝑥) = (1 + 2𝑥)2 (31) Then making a reflection about the y-axis results in the function 𝑔(𝑥) = ⋯ (a) −(1 + 2𝑥)2 (b) (1 − 2𝑥)2 (c) |1 + 2𝑥|2 (d) (1 + 2|𝑥|)2 (32) The graph of the function ℎ(𝑥) = (3 + 2𝑥)2 can be obtained from 𝑓(𝑥) by (a) Vertical shift -2 (b) Horizontal shift +2 (c) Horizontal shift -1 (d) Vertical shift +2 -------------------------------------------------------------------------------------------------------------------------- (33) Given the graph of the function 𝑓(𝑥) = 𝑥 2 , then the graph of 𝑔(𝑥) = 1 + (𝑥 − 2)2 can be obtained by (a) Horizontal shift +2, Vertical shift +1 (b) Horizontal shift -2, Vertical shift +1 (c) Horizontal shift -2, Vertical shift -1 (d) Horizontal shift +2, Vertical shift -1 -------------------------------------------------------------------------------------------------------------------------- 1 1 (34) Given the graph of the function 𝑥 then the graph of 4+𝑥 can be obtained by (a) Vertical shift –4 (b) Vertical shift +4 (c) Horizontal shift -4 (d) Horizontal shift +4 10  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 (35) The function 𝑔(𝑥) = −3 − √𝑥 − 3 can be obtained from the function 𝑓(𝑥) = 3 + √𝑥 − 3 by (a) Reflection about x-axis (b) Horizontal shift +3 (c) Reflection about y-axis (d) Vertical shift +6 ------------------------------------------------------------------------------------------------------------------------- (36) Given the graph of the function 𝑓(𝑥) = √−𝑥 , then the graph of 𝑔(𝑥) = √2 − 𝑥 can be obtained by (a) Horizontal shift -2 (b) Horizontal shift +2 (c) Vertical shift +2 (d) Reflection about y-axis -------------------------------------------------------------------------------------------------------------------------- (37) The graph of the function 𝑔(𝑥) = 𝑥 2 + 4𝑥 can be obtained from 𝑓(𝑥) = 𝑥 2 − 4 by (a) Vertical shift by 4 (b) Horizontal shift by 4 (c) Horizontal shift by 2 (d) Horizontal compression (e) Horizontal shift by -2 ------------------------------------------------------------------------------------------------------------------------ (38) Given 𝑓(𝑥) = |𝑥|, if 𝑓(𝑥) is reflected about x-axis and shifted 8 units up and 10 units to the right to generate the function 𝑔(𝑥). The function 𝑔(𝑥) = ⋯ (a) |𝑥 + 10| + 8 (b) −|𝑥 − 10| + 8 (c) −|𝑥 − 8| + 10 (d) |𝑥 + 8| − 10 ------------------------------------------------------------------------------------------------------------------------- (39) By reflecting the graph of the function 𝑓(𝑥) = 𝑥 2 + 2𝑥 + 1 about the y-axis, the resulting function is (a) 𝑥 2 − 2𝑥 − 1 (b) −𝑥 2 − 2𝑥 + 1 (c) (1 − 𝑥)2 (d) 𝑥 2 + 2𝑥 − 1 ------------------------------------------------------------------------------------------------------------------------- sin 𝑥 (40) For the function 𝑓(𝑥) = 1−cos 𝑥 , the domain is 𝜋 (a) 𝑅 − {2𝜋𝑛}, 𝑛 = 0, ±1, ±2, … (b) ]0,2[ (c) 𝑅 − { 2 𝑛} , 𝑛 = ±1, ±3, … (d) 𝑅 − {0} € 𝑅 − {𝜋𝑛}, 𝑛 = 0, ±1, ±2, … -------------------------------------------------------------------------------------------------------------------------- (41) The function 𝑓(𝑥) = tan √𝑎𝑥 , 𝑎 ≠ 0 is (a) General (b) Even (c) Odd (d) Depends on 𝑎 *) For 𝑓(𝑥) = (4 sin 𝑥 cos 𝑥)2 + 1 (42) The amplitude is (a) 3 (b) 4 (c) 16 (d) 2 11  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 (43) The period is 𝜋 𝜋 (a) 𝜋 (b) 2𝜋 (c) 2 (d) 4 -------------------------------------------------------------------------------------------------------------------------- (44) From the graph of the function 𝑓(𝑥) = | sin 𝑥 |, then the fundamental period of 𝑓(𝑥) is 𝜋 (a) 𝜋 (b) 2 𝜋 (c) 2 (d) Non-periodic (45) The range of 𝑦 = | tan 3𝑥 | is 𝑛𝜋 𝑛𝜋 (a) 𝑅 (b) [0, ∞[ (c) 𝑅 − { } , 𝑛 𝑜𝑑𝑑 (d) 𝑅 − { } , 𝑛 𝑜𝑑𝑑 2 6 -------------------------------------------------------------------------------------------------------------------------- *) For the function 𝑓(𝑥) = sin2 𝑥 + 2 cos2 𝑥, (46) The amplitude is 1 1 1 1 (a) 2 (b) 16 (c) 4 (d) 8 (47) The fundamental period is 3𝜋 𝜋 (a) (b) 2 (c) 𝜋 (d) 2𝜋 2 -------------------------------------------------------------------------------------------------------------------------- 𝜋𝑥 (48) For 𝑓(𝑥) = 5 cos(𝜋𝑥) sin(𝜋𝑥) + 3 tan( 2 ), the fundamental period is (a) 1 (b) Not periodic (c) 𝜋 (d) 2 -------------------------------------------------------------------------------------------------------------------------- √𝑥 2 −1 sin 𝑥 2 (49) For 𝑓(𝑥) = , the domain is 𝑥−1 (a) [−1,1[ (b) 𝑅−] − 1,1] (c) [0,1[ (d) ]1, ∞[ (e) ] − 1,1[ -------------------------------------------------------------------------------------------------------------------------- 𝜋 (50) If 𝑓 is an odd function, then cos(𝑓(𝑥) + 2 ) is (a) Even (b) General (c) Odd (d) cannot be determined ------------------------------------------------------------------------------------------------------------------------- 1 *) For 𝑓(𝑥) = (sin 𝑥 + cos 𝑥) √2 (51) The amplitude = 1 (a) √2 (b) (c) 0 (d) 1 √2 (52) This function is periodic and repeats itself every 𝜋 𝜋 (a) 2𝜋 (b) 𝜋 (c) 2 (d) 3 -------------------------------------------------------------------------------------------------------------------------- 12  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 53) The sketch of 𝑓(𝑥) − 1 = (𝑥 − 2)2 is a) b) c) d) -------------------------------------------------------------------------------------------------------------------------- 54) The sketch of 𝑓(𝑥) = 3 cos(𝜋 𝑥) is a) b) c) d) * ) For the function 𝑓(𝑥) = 2 cos 2 (2 𝑥) 55) The amplitude equals a) 1 b) 2 c) 0.5 d) √2 13  Faculty of Engineering Department of Engineering Mathematics and physics Fall 2024 56) The frequency equals 2 a) 𝜋 b) 2 𝜋 c) 0.5 𝜋 d) 𝜋 57) The sketch of 𝑓(𝑥) is a) b) c) d) -------------------------------------------------------------------------------------------------------------------------- (58) For the function 𝑓(𝑥) = √𝑥 2 + 1 and 𝑔(𝑥) = tan 𝑥, then 𝑓(𝑔(𝑥)) is: (a) can’t be determined (b) even (c) odd (d) general -------------------------------------------------------------------------------------------------------------------------- *) If 𝑓(𝑥) is an odd function and 𝑔(𝑥) is an even function, for ℎ(𝑥) = sin (𝑔(𝑓(𝑥))), ℎ(𝑥) (59) The function 3 is: (𝑓(𝑥)) (a) can’t be determined (b) even (c) odd (d) general 4 (60) The function tan ((𝑓(𝑥)) ) is: (a) can’t be determined (b) even (c) odd (d) general Ques. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Ans. c f b a a f a a e c f a a b b d b Ques. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Ans. b b c b d b c c b f e b d b c a c Ques. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Ans. a b e b c a a d c a b a c d b c d Ques. 52 53 54 55 56 57 58 59 60 Ans. a a a a d b b c b 14