Let f : [ 3 − π , 3 2 π ] → [ 0 , 4 ] be a function defined as f ( x ) = 3 sin x − cos x + 2 then f − 1 ( x ) is given by

Question

Let f : [ 3 − π ​ , 3 2 π ​ ] → [ 0 , 4 ] be a function defined as f ( x ) = 3 ​ sin x − cos x + 2 then f − 1 ( x ) is given by

TestHub · Accepted Answer

Since f is one-one onto, f is invertible. Now, f ∘ f − 1 ( x ) = x . 2 sin ( f − 1 ( x ) − 6 π ​ ) + 2 = x f − 1 ( x ) = sin − 1 ( 2 x ​ − 1 ) + 6 π ​ ( ∴ ​ 2 x ​ − 1 ​ ≤ 1∀ x ∈ [ 0 , 4 ] ) Also, using f − 1 ( x ) = 2 π ​ − cos − 1 2 x − 2 ​ + 6 π ​ = 3 2 π ​ − cos − 1 ( 2 x − 2 ​ )

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