Question

The inverse of the bijective function $f\left(x\right)=2\sqrt[3]{3x+1}-3,$ is

• A. ${\left(\frac{x+1}{2}\right)}^3+3$
• B. ${\left(\frac{x-3}{2}\right)}^3-1$
• C. ${\left(\frac{x+3}{2}\right)}^3-1$
• D. ${\left(\frac{x+1}{2}\right)}^3-3$

Answer 1

The correct option is C ${\left(\frac{x+3}{2}\right)}^3-1$

Let $y=f\left(x\right)$
$y=2\sqrt[3]{3x+1}-3$
switch $x$ and $y$
$\Rightarrow x=2\sqrt[3]{3y+1}-3\Rightarrow \frac{x+3}{2}=\sqrt[3]{3y+1}\Rightarrow y+1={\left(\frac{x+3}{2}\right)}^3\Rightarrow y={\left(\frac{x+3}{2}\right)}^3-1$
replace $y$ with $f^{-1}\left(x\right)$
$\Rightarrow f^{-1}\left(x\right)={\left(\frac{x+3}{2}\right)}^3-1$