JEE Main202125 Feb 2021Morning ShiftMathematicsFunctionsActual
Let f , g : N → N such that f n + 1 = f n + f 1 ∀ n ∈ N and g be any arbitrary function. Which of the following statements is NOT true?
Options
- AIf f is onto, then f n = n ∀ n ∈ N
- BIf g is onto, then f o g is one-one
- Cf is one-one
- DIf f o g is one-one, then g is one-one
Correct answer
B. If g is onto, then f o g is one-one
Step-by-step solution
f n + 1 - f n = f 1 A.P. with common difference = f ( 1 ) General term = T n   =   f ( 1 )   +   ( n - 1 ) f ( 1 )   =   n f ( 1 ) ⇒   f n = n f 1 ⇒ f is one-one Now, Let f g x 2 = f g x 1 ⇒   g x 2 = g x 1 (as f is one-one) ⇒   x 1 = x 2 (as f o g is one-one) ⇒   g is one-one For f to be onto, f ( n ) should take all the values of natural numbers. As f ( x ) is increasing, f ( 1 ) = 1 ⇒ f ( n ) = n Now, f g n = g n   f 1 may