Question

The inverse of the function$f\left(x\right)=(1-(x-5{)}^3{)}^{1/5}$is

• A. $5-(1-x^5{)}^{1/3}$
• B. $5+(1-x^5{)}^{1/3}$
• C. $5+(1+x^5{)}^{1/3}$
• D. $5-(1+x^5{)}^{1/3}$

Answer 1

The correct option is B $5+(1-x^5{)}^{1/3}$

Let $y=(1-(x-5{)}^3{)}^{\frac{1}{5}}$
$y^5=1-(x-5{)}^3$
$(x-5{)}^3=1-y^5$
$x-5=(1-y^5{)}^{\frac{1}{3}}$
$x=5+(1-y^5{)}^{\frac{1}{3}}$

Replacing $x$ with $y$ gives us

$f^{-1}\left(x\right)=5+(1-x^5{)}^{\frac{1}{3}}$