Question

Find the domain of f(x) = cos^-1 ( x² -3 ) .
1. [ -1, 1 ]
2. [ -3 , 3 ] 
3.  [ - \(\sqrt{2}\) , -2 ] ⋃ [ \(\sqrt{2}\) , 2 ]
4.  ( - \(\sqrt{2}\) , -2 ) ⋃ ( \(\sqrt{2}\) , 2 ) 

Answer 1

Correct Answer - Option 3 :  [ - \(\sqrt{2}\) , -2 ] ⋃ [ \(\sqrt{2}\) , 2 ]

Concept:

The domain and range of cos^-1x are [ -1, 1 ] and [ 0, π ]

Calculation:

The domain of  cos-1x is   [ -1 , 1] .

Let y = cos-1 ( x2 -3 )

⇒ cos y = x² - 3 

As we know -1 ≤ cos θ  ≤ 1

So, -1 ≤ cos y  ≤ 1

⇒ -1 ≤   x2 - 3 ≤ 1  

⇒ -1 + 3 ≤   x2 - 3 + 3 ≤ 1 + 3 

⇒ 2 ≤ x² ≤ 4 

⇒ x² ≥ 2 and x² ≤ 4

So, x ∈ [-∞ , -√2] ∪ [ √2, ∞] and x ∈ [-2, 2]

From combining the above results, we get 

⇒  -√2 ≤ x ≤ - 2  and √2 ≤ x ≤ 2 

∴  x ∈  [ -√2 , -2 ] ⋃ [ √2 , 2 ]

The correct option is 3