Determining the Graph, Domain, and Range of ln(arctan(x))

[dekoi]

f(x)=ln(arctan(x))

How does one determine the graph, domain, and range of the above?

 

[Jameson]

Start with domain, which in turn can help see the range. Look at the outer function...ln(x). For what kind of numbers is this defined? What sort of numbers are not in it's domain? And when you find that out, when does arctan(x) fall into those acceptable ranges?

 

[dekoi]

How does one determine the graph of arctan(x) ?

 

[Jameson]

The graph of arctan(x) is something you're going to have to be very familiar. The domain and range of it can be decuded using qualities of inverses. Let's say you have the function f(x), and it's inverse is g(x). Then for any point (a,b) on f(x) there is a corresponding point (b,a) on g(x). Also, the domain and range are opposites. The domain of f(x) is the range of g(x) and the range of f(x) is the domain of f(x). Now we have to restrict the x-values of the arctan(x) graph to maintain functionality. I'll give you a hint, from the tangent graph, pick the section from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$. Now from that, what's the domain and range of arctan(x)?