What is y equal to?

Calculation:

Given,

The function is \( y = \sin^{-1} \left( x - \frac{4x^3}{27} \right) \), and we are tasked with simplifying it.

\( y = \sin^{-1} \left( x - \frac{4x^3}{27} \right) \)

We recognize that we can factor the expression as:

\( x - \frac{4x^3}{27} = \left( \frac{x}{3} \right) \left( 3 - 4 \left( \frac{x}{3} \right)^3 \right) \)

Thus, the function becomes:

\( y = \sin^{-1} \left( \frac{x}{3} \left( 3 - 4 \left( \frac{x}{3} \right)^3 \right) \right) \)

Recognizing that this fits a known identity for inverse trigonometric functions:

\( \sin^{-1}(x) = 3 \sin^{-1} \left( \frac{x}{3} \right) \)

Thus, we can simplify the expression to:

\( y = 3 \sin^{-1} \left( \frac{x}{3} \right) \)

Hence, the correct answer is Option 4.