<s>[INST] Find the inverse of $f(x)= 3x^2 - 3x -11$, where $x>2$ 
Find the inverse of $f(x)= 3x^2 - 3x -11$, where $x > 2$

The answer guide says the answer is $f^{-1}(x)=\frac{1}{2} + \sqrt{ \frac{x}{3}} +\frac{47}{12}$,
but i had $\frac{1}{2} + \sqrt{\frac{x}{3} +\frac{47}{12}}$ instead.
How do i get the correct answer?
 [/INST] To find the innverse of $f(x)$, you have to swap $x$ and $y$, obtaining:
$$x=f(y)=3y^2-3y-11=\left(\sqrt{3}y-\frac{\sqrt{3}}{2}\right)^2-11-\frac{3}{4}$$
From here you solve for $y$, having:
$$x+11+\frac{3}{4}=3\left(y-\frac{1}{2}\right)^2$$
And so:
$$\sqrt{\frac{x}{3}+\frac{47}{12}}=y-\frac{1}{2} \leftrightarrow y = \sqrt{\frac{x}{3}+\frac{47}{12}}+\frac{1}{2}$$
Your answer is correct. There is certanly a typing error in the answer given.
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