JEE Main 2022 — Functions Question with Solution

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

\matrix{ {f(2) = 1,} & {g(2) = 3} & \Rightarrow & {(f + g)\,(2) = 4} \cr {f(3) = 1,} & {g(3) = 4} & \Rightarrow & {(f + g)\,(3) = 5} \cr {f(4) = 2,} & {g(4) = 5} & \Rightarrow & {(f + g)\,(4) = 7} \cr {f(5) = 1,} & {g(5) = 6} & \Rightarrow & {(f + g)\,(5) = 7} \cr }

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

$$\therefore$$ $$f+g$$ 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