# Relationship between inverse function, origin function and its domain - with an example

I am having problem with proper math-fashioned style of solving this task. It's task from my exam and apparently I didn't do well enough to get 2.5 out of 5 points.

Find inverse function $$f^{-1}$$to function $$f(x)= \arcsin{x^3}+\frac{\pi}{2}$$. Find the domain of $$f$$ and $$f^{-1}$$.

Step 1: domain of $$f$$:

$$-1 \leq x^3 \leq 1$$

$$\Rightarrow -1 \leq x^3 \land x^3\leq1$$

$$-1 \leq x \land x \leq 1$$

$$\Longrightarrow D_{f}=[-1;1]$$

Step 2: finding the inverse function.

$$f(x)=\arcsin{x^3} + \frac{\pi}{2}$$

$$y=\arcsin{x^3} + \frac{\pi}{2}$$

$$x \longleftrightarrow y$$

$$x=\arcsin{y^3} + \frac{\pi}{2}$$

$$x-\frac{\pi}{2}=\arcsin{y^3} \quad \quad /\sin(...) \quad \quad \quad \quad **HERE**$$

$$\sin{(x-\frac{\pi}{2})} = y^3 \quad \quad / ^{\sqrt{}}\quad \quad \quad \quad \quad \quad **HERE**$$

$$\sqrt{\sin{(x-\frac{\pi}{2})}} = y$$

$$\sqrt{\sin{(x-\frac{\pi}{2})}} = f^{-1}(x)$$

Question 1: in the parts where I wrote HERE the teacher on my paper wrote "assumptions!". (URL to my paper: https://i.imgur.com/zkJxAOD.jpg ) What does it mean, what kind of assumptions I should have made?

Question 2: What if I was multiplying by $$\log{(...)}$$, $$\ln{(...)}$$ or taking the even root like $$^{\sqrt{}}$$? What kind of assumptions should I make? Are there any other cases different than $$\arcsin$$, $$\arccos$$, $$\log$$, even root?

Step 3: Finding the domain of $$f^{-1}(x)$$:

Let's ignore it, but I have some questions here.

Because the initial function was $$\arcsin{x^3}$$ and $$\arcsin{\text{(...something...)}}$$ is based on $$[-\frac{\pi}{2}; \frac{\pi}{2}]$$ from domain of $$\sin{\text{(...something...)}}$$ (that's the only way, only in that interval sine function is injective, right?), then this is directly related to my $$Question 1$$?

Assumptions from that part are needed and directly related to the domain of $$f^{-1}(x)$$?

Check my calculations and paper please and tell me all mistakes I did, I need someone unforgiving to check what I wrote. Thanks