I'm a layman in mathematics so I apologize if I'm not using the correct mathematical notations.
I would like to map a unit square in $\mathbb{R}^2$ with corners in $(0,0)$ and $(1,1)$ to an area in $\mathbb{R}^2$ that is bounded by two parametrized functions.
Suppose I have the following two parametrized functions with $t,u\in[0,1]$ $$f_1(t,u)=\binom{t \cdot cos(\pi\cdot(1-u))}{t^2}$$ and $$f_2(t,u)=\binom{t\cdot(1-0.2u)+0.1u-0.5}{-(t-0.5)^4-0.5u+1}$$
These functions produce the two plots (for $f_1$ and $f_2$ respectively)
Both plots used $t\in[0,1]$ and $u_i=\frac{i}{k}$, where $i\in0..k$ and $k=8$ is the amount of discrete steps.
Now those two functions should somehow be combined to obtain the following parametrized area:
The problem is that I have no clue how I can do this or if it's even possible the way I've set this up. I also think the functions need to be adjusted somehow because right now they couldn't produce such an area.
Basically my question is this:
I need to insert some values $t,u$ (that are normalized to $[0,1]$) and get a $x$ and a $y$ which are inside a specific area bounded by functions. How could I achieve this? Or am I going into the completely wrong direction and this is a XY-Problem? Is there a better and simpler alternative then?