MHB Domain and range of inverse functions (circular and hyperbolic)

[Dethrone]

I've always been having trouble with the domain and range of inverse trigonometric functions. For example, let's start with an easy one: $\sin^{-1}\left({x}\right)$

Process: First, I draw out the function of $\sin\left({x}\right)$. Then I look at its range and attempt to restrict it so that it is invertible, which is from $\frac{-\pi}{2}$ to $\frac{\pi}{2}$. Now to draw the graph, I have to rotate it by the axis $y=x$...I can't seem to visualize how to do that...Do I turn my page sideways?

Let's try another one: $\sec^{-1}\left({x}\right)$
Drawing $\sec\left({x}\right)$ first, I look at its asymptotes which are when $\cos\left({x}\right)$ is equal to $0$ on... Does someone actually go through all this work to find the domain and range of $\sec^{-1}\left({x}\right)$? Or do they have it memorized?

This seems very tedious and a lot of work to know the domain and range of inverse functions. Is there a quicker way? (Wondering)

 

[I like Serena]

I've always been having trouble with the domain and range of inverse trigonometric functions. For example, let's start with an easy one: $\sin^{-1}\left({x}\right)$

Process: First, I draw out the function of $\sin\left({x}\right)$. Then I look at its range and attempt to restrict it so that it is invertible, which is from $\frac{-\pi}{2}$ to $\frac{\pi}{2}$.

Yep. That's a way to do it.

I usually visualize the unit circle in my mind and consider the range of the sine and the corresponding angles "that make sense".

Now to draw the graph, I have to rotate it by the axis $y=x$...I can't seem to visualize how to do that...Do I turn my page sideways?

Not rotate - flip.
That is, keep the page fixed at bottom-left and top-right, and flip it.

Or you start with the sine graph and the line $y=x$ and then mirror a couple of key points in the line.

Alternatively you can start by marking the horizontal axis with -1 and +1, and mark a couple of points to help draw the graph.

Let's try another one: $\sec^{-1}\left({x}\right)$
Drawing $\sec\left({x}\right)$ first, I look at its asymptotes which are when $\cos\left({x}\right)$ is equal to $0$ on... Does someone actually go through all this work to find the domain and range of $\sec^{-1}\left({x}\right)$? Or do they have it memorized?

I'm afraid I never use $\sec$, so I'm usually not interested in its domain or range.
If someone asks, I just have to look it up, or figure it out again.
I usually use $\frac{1}{\cos x}$ instead.

This seems very tedious and a lot of work to know the domain and range of inverse functions. Is there a quicker way? (Wondering)

With practice they tend to stick to mind. But it's not too bad if you forget. You just need a quick and reliable way to deduce them. (Sweating)