Converting between explicit function and parametric function

Question

Given an explicit function $y = f(x)$, how to convert it to the respective parametric functions $x = f_1(t)\; y = f_2(t)$?

Given parametric functions $x = f_1(t)\; y = f_2(t)$, how to obtain the respective implicit function $f(x,y) = 0$?

Answer 1

The first question has infinitely many solutions. An example would be to choose $f_1$ as the identity function, $f_2$ as $f$, and the parameter $t$ is just $x$, i.e. $t$ varies in the same domain as $x$ does.

The second question also has infinitely many solutions. Let the implicit function be denoted as $F$ to distinguish it from $f$. One solution would be as follows. Assume that $f_1$ has an inverse function $f_1^{-1}$ (otherwise the parametric representation doesn't have much sense), so that $t = f_1^{-1}(x)$. Then you can convert the parametric form $x=f_1(t)$, $y=f_2(t)$ to the explicit form $y=f_2(f_1^{-1}(x))$, and then to the implicit form $F(x,y)=0$ where $F$ is defined by $F(x,y) = y - f_2(f_1^{-1}(x)) $