First slide

Introduction to real valued functions

Question

If $f (x) = {Sin}^{- 1} {\{3 - {(x - 6)}^{4}\}}^{1 / 3}$ then $f^{- 1} (x) =$

Moderate

A

$6 + \sqrt[4]{3 + \sin^{3} x}$
B

$6 + \sqrt[4]{3 - \sin^{3} x}$
C

$6 + \sqrt[4]{3 + \sin x}$
D

$6 + \sqrt[4]{3 - \sin x}$

Solution

$f^{- 1} (x) = y \Rightarrow x = f (y) = \sin^{- 1} {(3 - {(y - 6)}^{4})}^{1 / 3} \sin y = (3 - {(y - 6)}^{4})^{\frac{1}{3}} \sin^{3} y = 3 - {(y - 6)}^{4} \Rightarrow y - 6 = {(3 - \sin^{3} y)}^{\frac{1}{4}} \Rightarrow y = 6 + {(3 - \sin^{3} y)}^{\frac{1}{4}}$
$\Rightarrow f^{- 1} (x) = 6 + \sqrt[4]{3 - \sin^{3} x} .$

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