# Example of a function with exactly two horizontal and vertical asymptotes, and an odd function with its absolute maximum value at exactly two points? [closed]

A while back a person posted this question on this site. I played with it for a while until I found (in my opinion) some pretty nice examples. A couple of minutes after I posted my solution the OP deleted the post, so I decided to ask-and-answer the question myself to share the examples.

## Questions:

(a) A function with exactly two horizontal asymptotes and exactly two vertical asymptotes, but is defined everywhere else on $$\mathbb{R}$$.

(b) A continuous function on $$\mathbb{R}$$ with $$f(2)=−3$$, $$f(−3)=2$$, and whose graph has no 𝑥-intercept.

(c) An everywhere continuous odd function on $$\mathbb{R}$$ which achieves its absolute maximum value at exactly two points.

a) A function that works is $$\frac{1}{x(x-1)} +\arctan(x)$$ since the first part gives you the vertical asymptotes and the $$\arctan(x)$$ part gives you the horizontal ones.
b) It's impossible. Notice that at $$x = -3$$ you are above the $$x$$-axis (at a height of $$2$$) and that at $$x=2$$ you are below the $$x$$-axis (at a height of $$-3$$). If the function is continuous you should be able to draw its graph without lifting your pencil. So can you draw a line from above the $$x$$-axis to below the $$x$$-axis without lifting your pencil and without crossing the $$x$$-axis?
c) An example that is continuous (and even differentiable) is $$f(x) = \begin{cases} e^{2\pi+x} -1, & \text{for }\ x \le -2\pi\\ \sin(x), & \text{for }\ -2\pi