Understanding Asymptotes: Vertical and Horizontal in Reciprocal Functions

[Atomos]

On my first IB test of the year. Since this is IB, rank matters immensely in determining your final standardized mark. That marks puts me tied in second with someone else.

The first question of the test asks you to sketch the reciprocal of a function f(x) and label any asymptotes and points of invarience. This function went from negative infinity, to 4, then back down to about -2, and the graph indicated that it continued on, never quite reaching -2.
Vertical asymptotes are easy enough; where it crosses 0. However, there are apparently 2 horizontal asymptotes, one at -1/2, and one at 0. The 0 makes sense, NO WHERE on the reciprical function does the graph cross 0; that is impossible. The -1/2 does not make sense. As I said, the ealrier part of the function comes all the way up from negative infinity with no breaks in the graph, therefore it DOES pass -2 and this the reciprocal does cross -1/2. I asked the teacher (this was at the end of class, so I did not have much time to make my point because I had to go to my next class) and he was saying something about local asymptotes, a concept that I was never introduced to. I was always taught that asymptotes were lines a function would never cross.

Am I wrong?

 

[shmoe]

I'm not entirely sure I understand what f was, but in any case a function can cross it's asymptotes. For example, 1/x-1/x^2 has a horizontal asymptote of y=0, yet it crosses this line at x=1.

 

[HallsofIvy]

I wouldn't "argue the mark" if I were you. You seem to have the wrong idea about "horizontal asymptotes". From what you say it seems clear to me that there is an asymptote at -1/2 but not necessarily 0.

And it is quite possible for a function to cross a horizontal asymptote. The only requirement for a horizontal asymptote is that the value of the function approach it as x goes to + or - infinity.

 

[Atomos]

Ok thanks. Upon further research, my definition of asymptote is incorrect. I was taught in earlier math courses that a function could not cross the asymptote.

 

[HallsofIvy]

The graph of a function cannot cross a vertical asymptote.