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If a function f : X → Y defined as f(x) = y is one-one and onto, then we can define a unique function g : Y → X such that g(y) = x, where x ∈ X and y = f(x), y ∈ Y. Function g is called the inverse of function f. The domain of sine function is R and function sine : R → R is neither one-one nor onto. The following graph shows the sine function. Let sine function be defined from set A to [– 1, 1] such that inverse of sine function exists, i.e., sin⁻¹ x is defined from [– 1, 1] to A. On the basis of the above information, answer the following questions :
Find the value of tan⁻¹(–1/√3) + cot⁻¹(1/√3) + tan⁻¹[sin(–π/2)].
Find the domain of the function f(x) = sin⁻¹(x² – 4). Also, find its range.
Assertion (A) : Domain of y = cos⁻¹(x) is [−1, 1]. Reason (R) : The range of the principal value branch of y = cos⁻¹(x) is [0, π] − {π/2}.
Assertion (A) : cos⁻¹(cos 13π/6) is equal to π/6. Reason (R) : The range of the principal value branch of the function y = cos⁻¹ x is [0, π].
Assertion (A) : The range of the function f(x) = 2 sin⁻¹ x + 3π/2, where x ∈ [−1, 1], is [π/2, 5π/2]. Reason (R) : The range of the principal value branch of sin⁻¹(x) is [0, π].
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