Question

Let $f,g:N-\{1\}\to N$ 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)

$f:N-\{1\}\to Nf(a)=\alpha$
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 ]