How can I define this type of function?

Question

I want to define this function:

It is not important that the function goes through any specific point; I just want a function that has a shape like this.

I think it is a polynomial function but I can't figure out how this function has to look like.

I also had an other idea which was to sum this two functions :

The left one should be $f(x) = \frac1x$ and the right one $g(x) = -x^2$ (moved a little bit up and to the right). But for the right one would also be important that for every $x^2 < 0$ $g(x) = 0$; is there any function in mathematics which does this? I don't know if this works but was the only idea I had so far.

Answer 1

This function appears to exhibit asymptotic behavior about the x and y axes. Although it may at first appear to resemble a polynomial, a rational function would probably be a better approximation. Note that this looks very similar to the graph of $f(x)=\frac{1}{x}$ with a small "bump" close to the x axis. Your approach to consider the addition of two functions is very good. We want to add a function to $f(x)$ that produces a small bump but adds a negligible amount everywhere else.

So essentially, we want to add a rational function in which the denominator can never be 0 to eliminate the possibility for assymptotes and which tends towards 0 for very high or low numbers. $g(x)=\frac{x^2}{x^2+1}$ gives this desired behavior. Since we want the bump to be at a particular spot we need only apply a few basic transformations to this function such that $g(x)=-\frac{(x-3)^2}{(x-3)^2+1}+1$

Then to yield the function you desire, it suffices simply to take the sum $f(x)+g(x)$. Here is a pictoral representation of the graph:

Answer 2

Following your idea, I have attached the picture of $\frac1x + 10 \exp(-(x-5)^2)$

You can try to tune the function of the form of

$$\frac{a}{x}+b\exp(-(x-c)^2)$$