Consider f: R₊→ [4, ∞] given by f(x) = x² + 4. Show that f is invertible with inverse of f given by , where R₊ is the set of all non-negative real numbers.

⇒ f: R₊→ [4, ∞] given by f(x) = x² + 4

One-one:

Let f(x) = f(y)

⇒ x² + 4 = y² + 4

⇒ x² = y²

So, x = y [as x = y ∈ R]

Therefore, f is a one-one function.

Onto:

For y ∈ [4, ∞), let y = x² + 4

⇒ x² = y – 4 ≥ 0 [as y ≥ 4]

Therefore, for any y ∈ R, there exists such that

So, f is onto.

Thus, f is one-one and onto and therefore, f^-1 exists.

Let us define g: [4, ∞) → R₊ by,

Now,

So, gof = fog = I_R+

Hence, f is invertible and the inverse of f is given by

Hence, proved.