Question

Let $f, g: N -\{1\} \rightarrow N$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^\alpha$ divides $a$, and $g(a)=a+1$, for all $a \in N -\{1\}$. Then, the function $f+ g$ is

• A. one-one but not onto
• B. onto but not one-one
• C. both one-one and onto
• D. neither one-one nor onto

Answer 1

Answer: (D)

$f : N -\{1\} \rightarrow N f ( a )=\alpha$
Where $\alpha$ is max of powers of prime $P$
such that $p ^\alpha$ divides $a$. Also $g ( a )= a +1$
$ \therefore f (2)=1 $
$ f (3)=1 $
$ f (4)=2 $
$ f (5)=1$
$g (2)=3 $
$ g (3)=4$
$ g (4)=5$
$g (5)=6$
$\Rightarrow f (2)+ g (2)=4 $
$ ( f (3)+ g (3))=5 $
$ f (4)+ g (4)=7$
$ f (5)+ g (5)=7 $
$ \therefore $ Many one }$f ( x )+ g ( x ) $ does not cotain $1 $
$ \Rightarrow $ into function
$ \therefore $ Ans. (D) [neither one-one nor onto ]