I am looking to describe two continuous functions. One of them is strictly increasing on the real line and one of them is strictly decreasing on the real line. I want to describe these functions in terms of non-exponential and non-trigonometric elementary functions.
I have this constraint because in my real analysis course these functions have not been introduced yet.
I wrote the problem off as easy but now I realize that I would not know how to solve it.
Consider $$f(x)=\frac{x}{1+|x|}$$
and move it up and down, and reflect it, to get two continuous monotone functions that don't cross each other.
$$f(x)=\begin{cases} 2-x & \quad \text{ in }(-\infty, 1] \\ 1/x & \quad \text{ in }[1,\infty) \end{cases}$$ $$ g(x)= \begin{cases}x-2 & \quad \text{ in }(-\infty, 1]\\ -1/x & \quad \text{ in } [1,\infty) \end{cases}$$
Take the lines $y=x+1, y=-x-1$ in $[0,\infty)$ and smooth them out in $(-\infty,0]$, so that they both approach $0$ ,from above, below respectively.
Take any two cdfs of continuous increasing r.v.s on $(-\infty,\infty)$ ($F_1, F_2$) that don't involve the functions you're trying to avoid. Then $F_1(x)$ and ($k-F_2(x)$) (for $k$ outside $(0,2)$) should work.
So for example $F_1 = \frac{1}{2}(1+\frac{x}{1+|x|})\,,$ for $-\infty<x<\infty$ would do.
Alternatively, getting away from distribution functions, something like $\sqrt{1+x^2/4}+x/2$ and its negative would work.
