The inverse of f(x)=5−(x−8)513 is

Let y=f(x)=5−(x−8)51/3. Theny3=5−(x−8)5⇒(x−8)5=5−y3⇒x=8+5−y31/5⇒f-1(x)=8+(5-x3)15

The inverse of f(x)=5−(x−8)513 is

Let y=f(x)=5−(x−8)51/3. Theny3=5−(x−8)5⇒(x−8)5=5−y3⇒x=8+5−y31/5⇒f-1(x)=8+(5-x3)15

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The inverse of $f (x) = {(5 - (x - 8)^{5})}^{\frac{1}{3}}$ is

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$5 - (x - 8)^{5}$

$8 + {(5 - x^{3})}^{1 / 5}$

$8 - {(5 - x^{3})}^{1 / 5}$

${(5 - (x - 8)^{1 / 5})}^{3}$

answer is B.

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Detailed Solution

Let $y = f (x) = {(5 - (x - 8)^{5})}^{1 / 3}$ . Then

$\begin{aligned} y^{3} = 5 - (x - 8)^{5} \Rightarrow (x - 8)^{5} = 5 - y^{3} \Rightarrow x = 8 + {(5 - y^{3})}^{1 / 5} \\ \Rightarrow f^{- 1} (x) = 8 + {(5 - x^{3})}^{\frac{1}{5}} \end{aligned}$

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The inverse of $f (x) = {(5 - (x - 8)^{5})}^{\frac{1}{3}}$ is