Let the functions f : - 1 , 1 → ℝ and g : - 1 , 1 → - 1 , 1 be defined by f x =   2 x - 1 + 2 x + 1   &   g x = x - x , where x denotes the greatest integer less than or equal to   x . Let f ∘ g : - 1 , 1 → ℝ be the

Question

Let the functions f : - 1 , 1 &#8594; &#8477;  and  g : - 1 , 1 &#8594; - 1 , 1 be defined by  f x = &#160; 2 x - 1 + 2 x + 1 &#160; &#38; &#160; g x = x - x , where x denotes the greatest integer less than or equal to &#160; x . Let f &#8728; g : - 1 , 1 &#8594; &#8477;  be the composite function defined by  f &#8728; g x = f g x . Suppose c  is the number of points in the interval  - 1 , 1  at which  f &#8728; g  is not continuous, and suppose d  is the number of points in the interval  - 1 , &#160; 1  at which  f &#8728; g  is not   differentiable. Then the value of  c + d is _____________

Quantrex Academy · Accepted Answer

Correct answer: A. A

f x = 2 x - 1 + 2 x + 1 = - 4 x , x - 1 2 2 , - 1 2 ≤ x ≤ 1 2 4 x , x > 1 2 g x = x f g x = - 4 g x , g x - 1 2 2 , - 1 2 ≤ g x ≤ 1 2 4 g x g x > 1 2 Graph of g x is f g x = 2 , x ∈ - 1 , - 1 2 ∪ 0 , 1 2 4 x + 1 x ∈ - 1 2 , 0 4 x x ∈ 1 2 , 1 c = 1 at x = 0 d = 3 at x = - 1 2 , 0 , 1 2 , ∴ c + d = 4 .

JEE Advanced Mathematics Question (2020) — Solution

Question

Options

Answer

Step-by-step solution