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Question
Find the domain of `f(x) = sin^-1 ((|x| - 2)/3) + cos^-1 ((1 - |x|)/4)`
Solution
`-1 ≤ sin^-1(x) ≤ 1`
`-1 ≤ (|x| - 2)/3 ≤ 1`
– 3 ≤ |x| – 2 ≤ 3
– 3 + 2 ≤ |x| ≤ 3 + 2
– 1 ≤ |x| ≤ 5
|x| ≤ 5
Since – 1 ≤ |x| is not possible
– 5 ≤ x ≤ 5 .......(1)
By the definitions
`-1 cos^-1(x) ≤ 1`
`-1 ≤ (1 - |x|)/4 ≤ 1`
– 4 ≤ 1 – |x| ≤ 4
– 5 ≤ – |x| ≤ 3
– 3 ≤ |x| ≤ 5
– 3 ≤ |x| is not possible
– 5 ≤ x ≤ 5 .......(2)
From 1 and 2 we get
Domain is x ∈ [– 5, 5]
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