Question

Consider f: R₊ → [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f⁻¹ of given f by, where R₊ is the set of all non-negative real numbers.

Answer 1

f: R₊ → [4, ∞) is given as f(x) = x² + 4.

One-one:

Let f(x) = f(y).

∴ f is a one-one function.

Onto:

For y ∈ [4, ∞), let y = x² + 4.

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

.

∴ f is onto.

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

Let us define g: [4, ∞) → R₊ by,

∴

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