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Question 8

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

Answer

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

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