$\text{Consider the function}\\
\\
f(x)=6x^3-3\\
\\
\text{let }y=f(x), \text{ so we have}\\
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y=6x^3-3\\
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\text{now in order to find the inverse of the function, first we interchange x and y}\\
\text{so we get}\\
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x=6y^3-3$

$\text{now we will solve for y again and that will give the required inverse function.}\\
\text{so first we add 3 both sides to get}\\
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x+3=6y^3\\
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\text{now divide both sides by 6 to get}\\
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\frac{x+3}{6}=y^3\\
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\text{now take cube root both sides to get}\\
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\sqrt[3]{\frac{x+3}{6}}=\sqrt[3]{y^3}$

$\Rightarrow \sqrt[3]{\frac{x+3}{6}}=y\\
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\Rightarrow y=\sqrt[3]{\frac{x+3}{6}}\\
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\text{hence the inverse of the function is: }\\
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f^{-1}(x)=\sqrt[3]{\frac{x+3}{6}}$

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