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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 :
A function f: [-4, 4] → [0, 4] is given by f(x) = √(16 - x²). Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = √7.
Draw the graph of f(x) = sin⁻¹ x, x ∈ [−1/√2, 1/√2]. Also, write range of f(x).
Find the domain of y = sin⁻¹(x² - 4).
Write the domain and range (principle value branch) of the following functions: f(x) = tan⁻¹ x
Find the domain of f(x) = cos⁻¹(1 – x²). Also, find its range.
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