Full solution

y=(\sin^{-1}(5x^{2}))^{3}

Identify Function Components: Identify the given function and its components. $\newline$ The function is $y = (\sin^{-1}(5x^2))^3$ , which means we need to find the cube of the arcsine (inverse sine) of $5x^2$ .
Understand Arcsine Domain: Understand the domain of the arcsine function. The arcsine function, $\sin^{-1}(x)$ , is defined for $-1 \leq x \leq 1$ . Therefore, the expression inside the arcsine, $5x^2$ , must also be within this range for the function to be real.
Determine Range of x: Determine the range of x for which the function is defined. $\newline$ Since $5x^2$ must be between $-1$ and $1$ , we can say that $-\frac{1}{5} \leq x^2 \leq \frac{1}{5}$ . However, since $x^2$ is always non-negative, the actual range of x is $0 \leq x^2 \leq \frac{1}{5}$ , which simplifies to $0 \leq x \leq \sqrt{\frac{1}{5}}$ .
Cube Arcsine Function: Cube the arcsine function. $\newline$ To cube the arcsine of $5x^2$ , we simply raise the entire expression $\sin^{-1}(5x^2)$ to the power of $3$ . This is a straightforward operation and does not involve any manipulation of the function itself.
Write Final Expression: Write the final expression. $\newline$ The final expression for $y$ is simply the cube of the arcsine of $5x^2$ , which is $y = (\sin^{-1}(5x^2))^3$ . There is no further simplification possible without specific values for $x$ .