Warning: This is kind of an open-ended question.
Often I have seen people interested in solving equations like $f(t)=g(t)$, where $f(t)$ is made up of trig functions and $g(t)$ is not. I was wondering if there have ever been studied special functions that map $g(t)$ (or its parameters) to solutions of $f(t)=g(t)$.
Here is a concrete example: For each $a\in[1,\infty)$ and each $x\in\mathbb{R}$ there is a unique solution (in $t$) to the equation $$\sin(t)=at-x.$$ We could therefore define a special function $\text{ss}_a:\mathbb{R}\to\mathbb{R}$ by letting each $\text{ss}_a(x)$ denote the unique solution to the above.
Question 1. Have any such special functions been studied before? I don't mean that it has to be exactly of the above form, but just something like it.
EDIT:
In the comments section there was some question as to $\text{ss}_a$ is well-defined, so here is a short proof. Write $\sin(t)=at-x$ as $x=at-\sin(t)$ Then $\frac{dx}{dt}=a-\cos(t)$ so that $x$ is bijective increasing as a function of $t$. Thus it has an inverse which is bijective increasing, and this inverse coincides with $\text{ss}_a$.
In fact, by the Inverse Function Theorem we have that $\text{ss}_a$ is continuously differentiable everywhere for $a>1$ with $\text{ss}'_a=1/(a-\cos(\text{ss}_a(x))$. In case $a=1$ it is continuously differentiable except at $x=2\pi n$, $n\in\mathbb{Z}$.