I finally solved it:
f(x) = (5 (x+1) (x-1))/((x+1) (x-4))
However, I'm not happy about it. I had to experiment a bit with a graphing calculator to see what happens to 1/x after applying several transformations to it, until I ended having ratios of quadratic functions that shared one factor with each other (point discontinuities; imatfaal's link to discontinuities helped me learn more about them). Among many other things, I need to learn, as soon as possible, what happens to rational functions when applying several transformation to them. No book that I have studied has touched on that subject, only on the graph transformations of quadratic, cubic, exponential, logarithmic, root, and trigonometric functions. (If anybody could suggest a page or a book that could teach me more about transformations of rational functions, I would be very grateful). I had to also use the answers to the problem to be able to figure out the mathematical mechanisms behind this beautiful problem. Nevertheless, I enjoyed it very much, and even more because I interacted with all of you. This is the very first time that I look for help in the internet, in a forum. The experience was very good, and I hope that I can keep interacting with all of you. I also hope that I can help other people who may not know something that I may.
Thank you