Question

Can you please help me answer this question correctly. I would really appreciate it!​

Answer 1

Answer:

$f(x) = \frac{2}{3} {x}^{3} - 1 \\ let \: {f}^{ - 1} (x) \: be \: m \\ m = \frac{1}{2 {x}^{3} - 3 } \\ m(2 {x}^{3} - 3) = 1 \\ 2 {x}^{3} = \frac{1}{m} + 3 \\ x = {( \frac{1}{2m} + \frac{3}{2} )}^{ \frac{1}{3} } \\ therefore \\ {f}^{ - 1} (x) = ( \frac{1}{2x} + \frac{2}{1} ) ^{ \frac{1}{3} } \\ = ( \frac{1 + 4x}{2x} ) ^{ \frac{1}{3} }$