Question

y = arctan x
I need to figure out the alternative notation, domain, and range? I am stuck right now and cant figure it out

Answer 1

Alternative notation? Maybe \(\tan^{-1}x\)?

Answer 2

you were correct with the alternative notation!

Answer 3

Everything else is fairly simple if you check the inverse function:
\[y=\arctan x~~\iff~~\tan y=x\]
\(\tan y\) is defined over the domain, \(\cdots\cup\left(-\dfrac{3\pi}{2},-\dfrac{\pi}{2}\right)\cup\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\cup\left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right)\cup\cdots\). Basically, tangent is defined for every two quadrants of the unit circle. In other words, it has asymptotes at \(y=\dfrac{(2k+1)\pi}{2}\) where \(k\) is any integer.

\(\tan y\) attains all real numbers, so the range will be \((-\infty,\infty)\).

The domain and range of one function are the range and domain of its inverse. However, a function has an inverse if and only if it is one-to-one; \(\tan y\) is not, so what you do is restrict the domain to \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\). You can pick any one interval, so long as you pick only one; this particular interval keeps it simple.

So if we restrict the domain, for \(y=\tan x\) we have
\[\text{domain:}~-\frac{\pi}{2}<x<\frac{\pi}{2}\\
\text{range:}~-\infty<y<\infty\]
which means the the inverse function, \(y=\arctan x\) has
\[\text{domain:}~-\infty<x<\infty\\
\text{range:}~-\frac{\pi}{2}<y<\frac{\pi}{2}\]

Answer 4

awesome your such a great help, I somewhat now understand it!, except cant figure how to enter this stuff into my online class lol