Relations and Functions · Question 14

JEE Main 2025 (Online) 4th April Evening ShiftMCQMedium📈 Topic in 35% papers+4−1

Let the domains of the functions f(x) = 4 3 7 (8 - 2 (x2 + 4x + 5 ) ) and g(x) = -1 ( 7x + 10x - 2 ) be ( , ) and [ , ], respectively. Then 2 + 2 + 2 + 2 is equal to :

Answer

(1) 15

Solution

For f: 4 3 7( ) is defined when 3 7 (8 - 2(x2 + 4x + 5) ) > 0

7 (8 - 2(x2 + 4x + 5) ) > 1

8 - 2(x2 + 4x + 5) > 7

2(x2 + 4x + 5) < 1

x2 + 4x + 5 < 2 (and x2 + 4x + 5 = (x + 2)2 + 1 > 0 always)

x2 + 4x + 3 < 0 (x + 1)(x + 3) < 0 -3 < x < -1

( , ) = (-3, -1)

For g: -1 7x + 10x - 2 1, x 2

7x + 10x - 2 1 6x + 12x - 2 0 -2 x < 2

7x + 10x - 2 -1 8x + 8x - 2 0 x -1 or x > 2

Domain of g = [-2, -1], i.e. = -2, = -1

2 + 2 + 2 + 2 = 9 + 1 + 4 + 1 = 15

Answer: (1)

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