I have some functions like $\cos(x)$ and $\sin(x)$ by definition. I know domain of the definition because it's provided is in my course, but when I try find the domain of a function based off of those functions, I have some problems. Specifically:
- Why is the domain of $\sin^{-1}(x)$ equal to $[-1,1]$?
- Why is the domain of $\tan^{-1}(x)$ equal to $\mathbb{R}$ ?
If $f\colon A\subset \mathbb{R} \to \mathbb{R}$ is an invertible function, then the domain of the inverse function $f^{-1}$ is the image $f(A)$ of $f$.
Example: We restrict the domain of the sine function to $[-\frac{\pi}{2},\frac{\pi}{2}]$, because on this interval the sine function is strictly increasing, and also because the sine attains a maximum and minimum in this interval. The image is $[-1,1]$. So the domain of the inverse $\arcsin$ is $[-1,1]$.
You can give a similar argument for $\tan$.