question_answer

Let N be the set of natural numbers and two functions f and g be defined as \[f,\text{ }g\text{ }:\text{ }N\to N\] such that \[f(n)\,=\,\left\{ \begin{matrix}    \frac{n+1}{2}  \\    \frac{n}{2}  \\ \end{matrix} \right.\,\,\,\begin{matrix}    ;\,\,\,\,if\,\,n\,\,is\,\,odd  \\    ;\,\,if\,\,n\,\,is\,\,given  \\ \end{matrix}\,\,\,\,;\,\,and\]\[g\left( n \right)=n-{{\left( -1 \right)}^{n}}\] . Then fog is [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

A) neither one-one nor onto

B) onto but not one-one

C) both one-one and onto

D) one-one but not onto

Correct Answer: B

Solution :

For f(g(n)) \[at,\text{ }n=1\] \[g\left( n \right)=1-{{\left( -1 \right)}^{1}}\] \[=\text{ }1-\left( -1 \right)=2\] \[f\left( 2 \right)=\frac{2}{2}=1\] at, \[n=2\] \[g\left( n \right)=2-{{\left( -1 \right)}^{2}}\] \[=\text{ }2-\left( 1 \right)=1\] \[f\left( 1 \right)=0\] at, \[n\,\,=\,\,3\] \[g\left( n \right)=3-{{\left( -1 \right)}^{3}}\] \[=\text{ }3-\left( -1 \right)=4\] \[f\left( 4 \right)=2\] at, \[n=4\] \[g\left( n \right)=4-{{\left( -1 \right)}^{4}}\] \[=4-\left( 1 \right)=3\] \[f\left( 3 \right)=1\] many one onto or onto but not one-one