1. Asymptotes and Rational Functions

This is more for my overall understanding than a specific calculus question...

When the asymptotes block off x & y intercepts, i.e Asymptotes y = 1 & x = 1, why graph anything on the outside of those asymptotes? The function has no intercepts, and from what I can tell, that function has no specific concavity values that I can derive.

Why is it there? what is it's purpose?

2. because if you plug points into the equation for that graph ( $f(x) = \frac{x+2}{x-1}$ for example ), one will get points on those 2 lines. I'm not sure if that answered your question though.