Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202325 Jan 2023Morning ShiftMathematicsFunctionsActual

Let f : 0 , 1 → ℝ be a function defined by f x = 1 1 - e - x , and g x = f - x - f x . Consider two statements (I) g is an increasing function in 0 , 1 (II) g is one-one in 0 , 1 Then,

Options

  1. AOnly (I) is true
  2. BOnly (II) is true
  3. CNeither (I) nor (II) is true
  4. DBoth (I) and (II) are true

Correct answer

D. Both (I) and (II) are true

Step-by-step solution

Given, f : 0 , 1 → ℝ be a function defined by f x = 1 1 - e - x , and g x = f - x - f x Now using g x = f - x - f x we get, g ( x ) = 1 1 - e x - e x e x - 1 ⇒ g x = 1 + e x 1 - e x Now differentiating the above function with respect to x we get, ⇒ g ' x = 2 e x 1 - e x 2 > 0 Since g ' x is always positive So,   g x is one-one in 0 , 1 And g x is increasing in 0 , 1

Practice Functions on Quantrex Academy →

More from Functions

Let N denote the set of all positive integers. Consider the sets A = 1, 2, 3, 4, 5 and B = 1, 2, 3, 4, 5, 6, 7 . Let S be the set of all functions f: A B such that f(2) 2 and f(4) 2026Let f:(1, ) R be a function defined as f(x) = x-1 x+1 . Let f^ i+1 (x) = f(f^i(x)) , i=1, 2, , 25 , where f^1(x)=f(x) . If g(x) + f²⁶(x) = 0 , x (1, ) , then the area of the region 2026Let f be a polynomial function such that ₂(f(x)) = ( ₂ (2+ 2 3 + 2 9 + ) ) ₃ (1+ f(x) f(1/x) ) , x>0 and f(6)=37 . Then _ n=1 ¹⁰f(n) is equal to ________. 2026Let f: R R be defined as f(x) = 2x^2 - 3x + 2 3x^2 + x + 3 . Then f is : 2026Let e be the base of natural logarithm and let f: 1, 2, 3, 4 1, e, e^2, e^3 and g: 1, e, e^2, e^3 1, 1 2 , 1 3 , 1 4 be two bijective functions such that f is strictly decreasing a 2026Let f: R R be a function such that f(x) + 3f ( 2 - x ) = x , x R . Let the maximum value of f on R be . If the area of the region bounded by the curves g(x) = x^2 and h(x) = x^3 , 2026The sum of all the integral values of p such that the equation 3 ^2 x + 12 x - 3 = p , x R , has at least one solution, is: 2026Let A = 1, 2, 3, 4, 5, 6 . The number of one-one functions f: A A such that f(1) 3 , f(3) 4 and f(2) + f(3) = 5 , is __________. 2026 Full Functions list All JEE Main PYQs