Give examples of two functions f : N → N and g : N → N such that g

1.	Give examples of two functions f : N → N and g : N → N such that gof is onto but f is not onto. (Hint: Consider f (x) = x + 1 and \(g(x) =\begin{cases}x - 1 &if \; x > 1\\1 & if \; x = 1\end{cases}\)
Answer» f(x) = x+ lxl + 1 ≥ 1 + 1 ∀ x ∈ N (∵ v ≥ 1 x ∈ N) ⇒ f(x) ≥ 2 ∀ x ∈ N . R_f^ N as 1 t R_f Hence f is not onto gof : → ∼ → N such that gof (x) = g (f (x)) = g (x + 1) = (x+1) -1 [ ∵ ∀ x ∈ N,x+ 1 > 1] ⇒ gof (x) = x V x ∈ N ∴ Range of gof = N as gof is the identity f Hence gof is onto.

Give examples of two functions f : N → N and g : N → N such that gof is onto but f is not onto. (Hint: Consider f (x) = x + 1 and \(g(x) =\begin{cases}x - 1 &if \; x > 1\\1 & if \; x = 1\end{cases}\)

Answer»

f(x) = x+ lxl + 1 ≥ 1 + 1 ∀ x ∈ N

(∵ v ≥ 1 x ∈ N)

⇒ f(x) ≥ 2 ∀ x ∈ N . R_f^ N as 1 t R_f

Hence f is not onto

gof : → ∼ → N such that

gof (x) = g (f (x)) = g (x + 1) = (x+1) -1

[ ∵ ∀ x ∈ N,x+ 1 > 1]

⇒ gof (x) = x V x ∈ N

∴ Range of gof = N as gof is the identity f

Hence gof is onto.