The question states,
Find the inverse function of $y = g(x) = 6 x^3 + 7$, $g^{-1}(y) =?$
I have tried setting the equation to $y$ and then solving for $x.$ This resulted in the answer $\dfrac{(x-7)^{1/3}}{6}$. This answer is incorrect. I have also tried $\dfrac{(36(x-7))^{1/3}}{6}$ and just re-writing the equation as $7+6y^3.$ None of these answers are correct either. Please help.
You are close. Solving for $x$ in $y=6x^3+7$ goes like this: $$y=6x^3+7 \implies y-7=6x^3 \implies \frac{y-7}{6} = x^3 \implies \left(\frac{y-7}{6}\right)^{1/3} = x.$$