# Graphing the function $f(x) = \frac{(x-1)(x-2)}{(x(x+2)(x-3))}$

When I (try to) graph the function $f(x) = \frac{(x-1)(x-2)}{(x(x+2)(x-3))}$ I start by finding the vertical asymptotes $x = 0$, $x = -2$ and $x = 3$ and the roots $x = 1$ and $x = 2$. At this point I'm pretty confused about what happens next.

In class, we've been graphing functions that I feel are similar to $f(x)$ and when graphing those I proceed to find the horizontal asymptote and go from there but something doesn't feel right about this particular function.

This is very hard for me to put into words, I'm sorry about that.

• Also look at the limits, what happens when $x\rightarrow \infty$ or $x\rightarrow -\infty$ and look at the intersection with the $x$ and $y$-axis and use the derivative to find the extrema, and when the function is increasing or decreasing. This would give some more insight. – user112167 Nov 29 '13 at 17:20
• For large $|x|$, $f(x)$ 'behaves' like ${1 \over x}$. – copper.hat Nov 29 '13 at 17:23
• Find the intervals where the function is increasing and decreasing, as well as the critical points and the value of the function at those points – GTX OC Nov 29 '13 at 17:23
• @user112167 the limit of f(x) as x tends to infinity is 0 right? I think I understand what that means with one vertical asymptote but I have three so to the right of x = 3, y -> 0 as x -> infinity and to the left of x = -2, y -> negative infinity as x -> -2 I think? But happens in the intervals between? – LampsCan'tUseBikes Nov 29 '13 at 17:36
• @LampsCan'tUseBikes: You can also probe the cases where $x\to 2^+$ or $x\to 2^-$. – mrs Nov 29 '13 at 18:24

Some good point:

• Find the domain of the function. Here, $D_f=\mathbb R-\{0,+3,-2\}$ (Why?)
• Find the points where the $f$ meets the axes if it is possible. Here, they are $(1,0),(2,0)$.
• Find $L=\lim_{x\to\pm\infty}f(x)$. Here, $L$ is $0$ (Why?).
• Find the probable locally extreme by checking the points where $f'$ does not exist or $f'=0$.
• Find asymptotes, vertical, horizontal or inclined ones if exists.
• Make some additional points in which we can make the whole plot better.

• I am having a good breakfast and going to go to the university. :) – mrs Nov 30 '13 at 2:49
• @Sami I'm seeing you almost as often as I'm seeing dear Babak! And any friend of Babak's is a friend of mine! ;-) – Namaste Nov 30 '13 at 15:37
• Haaah! I knew there was some mathematical logic in friendship! ;-) – Namaste Nov 30 '13 at 15:54
• @amWhy: I saw them hours ago. I am so sorry for the one who did it unanimously. It's not fair. We all claim that we know logic. Is this logic??!! I don't think so. Stand with you. :-) – mrs Nov 30 '13 at 17:06
• @SamiBenRomdhane: We have been connected :) – mrs Nov 30 '13 at 17:09