Question

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

• A. $6+\sqrt[4]{43+{\sin }^{3}x}$
• B. $6+\sqrt[4]{43-{\sin }^{3}x}$
• C. $6+\sqrt[4]{43+\sin x}$
• D. $6+\sqrt[4]{43-\sin x}$

Answer 1

Answer: (B)

$6+\sqrt[4]{43-{\sin }^{3}x}$

Let $f\left(x\right)=y={\sin }^{-1}(3-(x-6{)}^4{)}^{\frac{1}{3}}$
$\sin y=(3-(x-6{)}^4{)}^{\frac{1}{3}}$
${\sin }^{3}y=3-(x-6{)}^4$
$(x-6{)}^4=3-{\sin }^{3}y$
$x-6=(3-{\sin }^{3}y{)}^{\frac{-1}{4}}$
$x=(3-{\sin }^{3}y{)}^{\frac{1}{4}}+6$
$x=f^{-1}\left(y\right)=(3-{\sin }^{3}y{)}^{\frac{1}{4}}+6.$