Question

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

• A. $5-(x-8{)}^5$
  
• B. $8+(5-x^3{)}^{\frac{1}{5}}$
  
• C. $8-(5-x^3{)}^{\frac{1}{5}}$
  
• D. $(5-(x-8{)}^{\frac{1}{5}}{)}^3$

Answer 1

The correct option is B $8+(5-x^3{)}^{\frac{1}{5}}$


Let $y=f\left(x\right)={(5-(x-8{)}^5)}^{\frac{1}{3}}$, then
$y^3=5-(x-8{)}^5\Rightarrow (x-8{)}^5=5-y^3$
$\Rightarrow x=8+(5-y^3{)}^{\frac{1}{5}}$
Let, $z=g\left(x\right)=8+(5-x^3{)}^{\frac{1}{5}}$
Now to check -
$f\left(g\right(x\left)\right)={[5-(x-8{)}^5]}^{\frac{1}{3}}$
$={(5-{\left[\right(5-x^3{)}^{\frac{1}{5}}]}^5)}^{\frac{1}{3}}=(5-5+x^3{)}^{\frac{1}{3}}=x$
Similarly, we can show that $f\left(f\right(x\left)\right)=x$.
Hence, $g\left(x\right)=8+(5-x^3{)}^{\frac{1}{5}}$ is the inverse of f(x).