Question

Let Y = {n^2 : n ∈ N} ⊂ N. Consider f : N → Y as f(n) = n2. Show that f is invertible. Find the inverse of f

Answer 1

f(n)=n²
so let y = n²
n = +√y or - √y
f ⁻¹(n) =  +√y or - √y

Answer 2

First ,let us understand what is invertible function.

$\large{\bullet{\mathfrak{\underline{Invertible\: Function:-}}}}$

A function f: X→ Y is defined to be invertible, if there exists a function g: Y→ X suxh that gof= Ix and Iy. The function g is called the inverse of f and is denoted by f^-1.

Now, coming to question

$\large{\bullet{\mathfrak{\underline{Proof:-}}}}$

let an arbitrary element y in Y is of the form n² , for some n€ N.

so,

→ n= √y

Now, another function g : Y→ N, defined g(y)= √y.

So,

gof(n) = g(n²)= √n²= n

and

fog(y) = f(√y)=(√y)²= y

which shows that

gof(n)= In and fog= Iy

•°• f is invertible with f^{-1} = g