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The domain of is

We need to find the domain of cos^{-1} (x^{2} – 4).

We must understand that, the domain of definition of a function is the set of "input" or argument values for which the function is defined.

We know that, domain of an inverse cosine function, cos^{-1} x is,

x ∈ [-1, 1]

Then,

(x^{2} – 4) ∈ [-1, 1]

Or,

-1 ≤ x^{2} – 4 ≤ 1

Adding 4 on all sides of the inequality,

-1 + 4 ≤ x^{2} – 4 + 4 ≤ 1 + 4

⇒ 3 ≤ x^{2} ≤ 5

Now, since x has a power of 2, so if we take square roots on all sides of the inequality then the result would be

⇒ ±√3 ≤ x ≤ ±√5

But this obviously isn’t continuous.

So, we can write as

1