Let g: N → N be defined as g ( 3n + 1 ) = 3n + 2 g ( 3n + 2 ) = 3n + 3 g ( 3n + 3 ) = 3n + 1 , for all n ≥ 0 Then which of the following statements is true ?

Correct answer: A. There exists an onto function f: N → N such that fog = f g: N → N g ( 3n + 1 ) = 3n + 2 g ( 3n + 2 ) = 3n + 3 g ( 3n + 3 ) = 3n + 1 g(x) = \ array ll x + 1; x = 3k + 1 x + 1; & x = 3k + 2 x - 2; & x = 3k + 3 array ) g(g(x)) = \ array ll x + 2; x = 3k + 1 x - 1; & x = 3k + 2 x - 1; & x = 3k + 3 array ) g(g(g(x))) = \ cases x; & x = 3k + 1 x; & x = 3k + 2 x; & x = 3k + 3 cases ) If f: N → N, f is a one-one function such that f ( g ( x ) ) = f ( x ) ⇒ g ( x ) = x , which is not the case If f: N → N, f is an onto function such that f ( g ( x ) ) = f ( x ) one possibility is f(x) = \ cases n; & x = 3n + 1 n; & x = 3n + 2 n; & x = 3n + 3 cases ; n ∈ N Here f(x) is onto, also f ( g ( x ) ) = f ( x ) x ∈ N

JEE Main 202125 JulyMorning Shift

Exam: JEE Main
Year: 2021
Date: 25 July
Shift: Morning Shift
Subject: Mathematics
Chapter: Functions

JEE Main 2021 · 25 July · Morning Shift · Mathematics · Functions

Let g: N → N be defined as g ( 3n + 1 ) = 3n + 2 g ( 3n + 2 ) = 3n...

Let $g: N \to N$ be defined as

$g ( 3n + 1 ) = 3n + 2$

$g ( 3n + 2 ) = 3n + 3$

$g ( 3n + 3 ) = 3n + 1$ , for all $n \ge 0$

Then which of the following statements is true ?

Options

AThere exists an onto function $f: N \to N$ such that $fog = f$
BThere exists a one-one function $f: N \to N$ such
that $fog = f$
C $gogog = g$
DThere exists a function $f: N \to N$ such that $gof = f$

Correct answer

A. There exists an onto function $f: N \to N$ such that $fog = f$

Step-by-step solution

g: N \to N $$ g ( 3n + 1 ) = 3n + 2 $$ g ( 3n + 2 ) = 3n + 3 $$ g ( 3n + 3 ) = 3n + 1 $$ g\left(x\right) = \left\{ $\begin{array}{ll}x + 1; x = 3k + 1 \\ x + 1; & x = 3k + 2 \\ x - 2; & x = 3k + 3\end{array}$ \right)

$g\left(g\left(x\right)\right) = \left\{$ $\begin{array}{ll}x + 2; x = 3k + 1 \\ x - 1; & x = 3k + 2 \\ x - 1; & x = 3k + 3\end{array}$ $\right)$ g\left(g\left(g\left(x\right)\right)\right) = \left\{ $\begin{cases}x; & x = 3k + 1 \\ x; & x = 3k + 2 \\ x; & x = 3k + 3\end{cases}$ \right)

If $f: N \to N, f$ is a one-one function such that $f ( g ( x ) ) = f ( x ) ⇒ g ( x ) = x$ , which is not the case

If $f: N \to N, f$ is an onto function such that $f ( g ( x ) ) = f ( x )$ one possibility is

$f\left(x\right) = \left\{$ \begin{cases}n; & x = 3n + 1 \\ n; & x = 3n + 2 \\ n; & x = 3n + 3\end{cases} $\right; n \in N$

Here $f\left(x\right)$ is onto, also $f ( g ( x ) ) = f ( x ) \forall x \in N$

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… JEE Advanced 2026 Let f:(1,∞)→ℝ be a function defined as f(x) = (x-1)/(x+1) . Let f^ i+1 (x) = f(f^i(x))… JEE Main 2026 Let f be a polynomial function such that _2(f(x)) = ( _2(2+(2)/(3)+(2)/(9)+…∞))·… JEE Main 2026 Let f: ℝ arrow ℝ be defined as f(x) = (2x^2 - 3x + 2)/(3x^2 + x + 3) . Then f is : JEE Main 2026 Let e be the base of natural logarithm and let f: \ 1, 2, 3, 4\ arrow \ 1, e, e^2, e^3\… JEE Main 2026 Let f: R arrow R be a function such that f(x) + 3f((π)/(2) - x) = x , x ∈ R . Let the… JEE Main 2026 The sum of all the integral values of p such that the equation 3 ^2 x + 12 x - 3 = p , x… JEE Main 2026 Let A = \ 1, 2, 3, 4, 5, 6\ . The number of one-one functions f: A arrow A such that f(1)… JEE Main 2026 For the function f:[1,∞) arrow [1,∞) defined by f(x)=(x-1)^4+1 , among the two… JEE Main 2026 Let for some α ∈ ℝ , f:ℝarrowℝ be a function satisfying f(x+y)=f(x)+2y^2+y+α xy for all… JEE Main 2026 Let [·] denote the greatest integer function. If the domain of the function f(x) = ^ -1… JEE Main 2026 The number of functions f: \ 1, 2, 3, 4\ arrow \ a, b, c\ , which are not onto, is: JEE Main 2026 If the domain of the function f(x) = _ (0.6) (|(2x-5)/(x^2-4)|) is (-∞, a] \ b\ [c, d)… JEE Main 2026 Given below are two statements: Statement I: The function f: R arrow R defined by… JEE Main 2026 The sum of all the elements in the range of f(x)=Sgn( x)+Sgn( x)+Sgn( x)+Sgn( x) , x ≠ n… JEE Main 2026 If g(x)=3 x^ 2 +2 x-3, f(0)=-3 and 4 g(f(x))=3 x^ 2 -32 x+72 , then f(g(2)) is equal to: JEE Main 2026 Full Functions list Functions JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs