Hi there!
[tex]\large\boxed{f^{-1}(x) = \sqrt[3]{\frac{x+4}{9} } }[/tex]
[tex]f(x) = 9x^{3} - 4[/tex]
Find the inverse by replacing f(x) with y and swapping the x and y variables:
[tex]x = 9y^{3} - 4[/tex]
Isolate y by adding 4 to both sides:
[tex]x + 4 = 9y^{3}[/tex]
Divide both sides by 9:
[tex]\frac{x+4}{9}= y^{3}[/tex]
Take the cube root of both sides:
[tex]y = \sqrt[3]{\frac{x+4}{9} }\\\\f^{-1}(x) = \sqrt[3]{\frac{x+4}{9} }[/tex]
