I'm just some random bum calculus II student who was tinkering around with calculus and I think I came up with some new function on my own. so the tl;dr of my function is that you take some function and project that function onto another one. The function is called $b(f(x),g(x))$ where $f(x)$ is the function, and $g(x)$ will be treated as the axis. Here are some picture examples.
$b(sin(.5x),5sin(.1x))$
$b(sin(x),.01x^2)$
If it turns out that this is something new, where should I send it to to spread the word? Thank you!
Expounding on my comment, it's not actually too hard to give a fairly explicit set of (messy) expressions for the curve you are describing. To see how this can be done, consider the notion of parallel curves described in at https://en.wikipedia.org/wiki/Parallel_curve. Here, what you basically do is move a fixed distance $d$ along the normal line at each point of the initial curve (your $g(x)$) to obtain the corresponding point on the parallel curve. This can be calculated based on the derivative of $g$, as the slope of the normal line is related to that of the tangent line (they differ by 90 degrees, or $\pi / 2$ radians). What you have done is to replace the constant $d$ with a function $f$ of the same $x$ as used in in $g$, hence the distance is now also a function of where you are on the $g$ function (so now, $d=f(x)$). It would not be too hard to give (messy) parametric equations for your $b$ function, which btw is generally actually a pair of functions, $b_x(x,f,g)$ (this returns the $x$-coordinates for points on the $b$ curve parametrically in terms of original $x$ values used in the $g$ function) and $b_y(x,f,g)$ (likewise for the $y$-coordinates, based on and same $x$), of one variable and 2 functions (of that same single variable).