Function $f(x)$ similar to exp(x) where $-f(x)$ is approximately $f(-x)$ I am wondering if there is a function $f(x)$ "similar" to the exponential function $\exp(x)$ such that:
$-f(x) \approx f(-x)$
I would also like $f(x)$ to have the following property:
$\frac{{f(a)}}{{f(b)}} = f(a - b)$
Or alternately,
$\frac{{f(a)}}{{f(b)}} \approx f(a - b)$

You might be interested in the hyperbolic sine "sinh". It is antisymmetric and its asymptotic behaviour for $x\to\infty$ is similar to the one of the exponential function.
