QJEEM22Maca26: JEE Main 2022 Mathematics - Class 11 JEEM22Maca26 Solution: Let be functions defined by , where is the maximum of the…

$f,g:N - \{ 1\} \to N$ defined as

$f(a) = \alpha$ , where $\alpha$ is the maximum power of those primes p such that p ^$\alpha$ divides a.

$g(a) = a + 1$ ,

Now,

$$\begin{array}{cccc}f(2)=1,& g(2)=3& \Rightarrow & (f+g)(2)=4\\ f(3)=1,& g(3)=4& \Rightarrow & (f+g)(3)=5\\ f(4)=2,& g(4)=5& \Rightarrow & (f+g)(4)=7\\ f(5)=1,& g(5)=6& \Rightarrow & (f+g)(5)=7\end{array}$$

$\because$ $(f + g)\,(5) = (f + g)\,(4)$

$\therefore$ $f + g$ is not one-one

Now, $\because$ ${f_{\min }} = 1,\,{g_{\min }} = 3$

So, there does not exist any $x \in N - \{ 1\}$ such that $(f + g)(x) = 1,2,3$

$\therefore$ $f + g$ is not onto