Question

If Y = {n²: n ∈ N} ⊂ N and the function f: N → Y as f(n) = n². Show that f is invertible. Also find the inverse of f.

Answer 1

Calculate: g: Y → N

Given: f(n) = n²

Let f(n) = y.

⇒ n² = y

⇒ n = ±√y.  Since f: N → N, n ∈ N

So, n is positive.

∴ n = √y. Let g(y) = √y

where g: Y → N

Now, f(n) = n² & g(y) = √y

Solve for prove gof = I_N.

gof = g(f(n))

⇒ gof = g(n²)

⇒ gof = \(\sqrt{(n^2)}\)

⇒ gof = n

Hence, gof = n = I_N   ...(1)

Solve for prove fog = I_Y.

fog = f(g(y))

⇒ fog = f(√y)

⇒ fog = (√y)² = y

Hence, fog(y) = y = I_Y  ...(2)

From (1) and (2),

gof = I_N and fog = I_Y

So, f is invertible and

Inverse of f = g(y) = √y.