2. Given the function $f(x)=-3(x+1)^{2}-2$ :
a) Determine the equation of the inverse of the $f(x)$

Step 1 ：We are given the function \(f(x)=-3(x+1)^{2}-2\)

Step 2 ：To find the inverse of a function, we swap the x and y (or f(x)) values in the original equation. This means we replace every x in the equation with y and solve for y.

Step 3 ：So, we get the equation \(x = -3(y + 1)^{2} - 2\)

Step 4 ：Solving this equation for y, we get two solutions: \(y = -\frac{7}{6} - \frac{\sqrt{11}i}{6}\) and \(y = -\frac{7}{6} + \frac{\sqrt{11}i}{6}\)

Step 5 ：These solutions are complex, which means they have imaginary parts. This is because the original function is a downward-opening parabola, which does not pass the horizontal line test, and thus does not have a real-valued inverse.

Step 6 ：\(\boxed{\text{The inverse of the function } f(x)=-3(x+1)^{2}-2 \text{ is } f^{-1}(x) = -\frac{7}{6} - \frac{\sqrt{11}i}{6}, -\frac{7}{6} + \frac{\sqrt{11}i}{6}}\)