I'm searching for a monotonically increasing function, defined for all reals, that is concave down and a "gentle curve" (no aymptotes).

Question

I'm searching for a monotonically increasing function, defined for all reals, that is concave down and a "gentle curve" (no aymptotes). The link below provides an image:

example of gentle concave down curve

This post has some examples of near misses: Is there a bijective, monotonically increasing, strictly concave function from the reals, to the reals?

For instance: y = -e^(-x)

However, this is not by any means a gentle curve, outside of the domain (-3,3). The rest of the graph looks essentially like a right angle.

Answer 1

Perhaps try something like $$ f(x) := -\frac{x^2}{4}+\frac{x\sqrt{x^2+4}}{4}+\operatorname{asinh}\frac{x}{2} \tag1$$

We have $$ \lim_{x\to\infty} f(x) = \infty,\quad \lim_{x\to\infty} f'(x) = 0, \\ \lim_{x\to-\infty} f(x) = -\infty,\quad \lim_{x\to-\infty} f'(x) = -\infty, $$

To find this, I started with the hyperbola $y(y+x)=1$ with asymptotes $y=0$ to the right and $y=-x$ to the left. But you said you did not want asymptotes. So I integrated it, $$ \int \left(-\frac{x}{2}+\frac{\sqrt{x^2+4}}{2}\right) \,dx $$ to get $(1)$.