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?
Solution
g : N $\to$ N <br><br>g(3n + 1) = 3n + 2,<br><br>g(3n + 2) = 3n + 3,<br><br>g(3n + 3) = 3n + 1<br><br>$$g(x) = \left[ {\matrix{
{x + 1} & {x = 3k + 1} \cr
{x + 1} & {x = 3k + 2} \cr
{x - 2} & {x = 3k + 3} \cr
} } \right.$$<br><br>$$g\left( {g(x)} \right) = \left[ {\matrix{
{x + 2} & {x = 3k + 1} \cr
{x - 1} & {x = 3k + 2} \cr
{x - 1} & {x = 3k + 3} \cr
} } \right.$$<br><br>$$g\left( {g\left( {g\left( x \right)} \right)} \right) = \left[ {\matrix{
x & {x = 3k + 1} \cr
x & {x = 3k + 2} \cr
x & {x = 3k + 3} \cr
} } \right.$$<br><br>If f : N $\to$ N, if is a one-one function such that f(g(x)) = f(x) $\Rightarrow$ g(x) = x, which is not the case<br><br>If f : N $\to$ N f is an onto function<br><br>such that f(g(x)) = f(x),<br><br>one possibility is <br><br>$$f(x) = \left[ {\matrix{
x & {x = 3n + 1} \cr
x & {x = 3n + 2} \cr
x & {x = 3n + 3} \cr
} } \right.$$ n$\in$N<sub>0</sub><br><br>Here f(x) is onto, also f(g(x)) = f(x) $\forall$ x$\in$N
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
This question is part of PrepWiser's free JEE Main question bank. 195 more solved questions on Sets, Relations and Functions are available — start with the harder ones if your accuracy is >70%.