Answer
$f^{-1}(x)=\sqrt[3]{x+3}$
Work Step by Step
The given function, $
f(x)=x^3-3
,$ is equivalent to
\begin{array}{l}\require{cancel}
y=x^3-3
.\end{array}
Interchanging the $x$ and $y$ variables and then solving for $y$ result to
\begin{array}{l}\require{cancel}
x=y^3-3
\\\\
x+3=y^3
\\\\
\sqrt[3]{x+3}=y
\\\\
y=\sqrt[3]{x+3}
.\end{array}
Hence, the inverse is $
f^{-1}(x)=\sqrt[3]{x+3}
.$
In the graph above, the black graph is the graph of $f(x)$ and the red graph is the graph of $f^{-1}(x).$
