$$ f(x,k) = y$$ $$x<=k$$ $$x,k \in R$$ $$y \in [0,1]$$
The domain of the function would be $x$ to $k$ and the range would be $0$ to $1$. such that for $x=x$ the $y=0.0$ and for $x=k$ , $y=1.0$, the value of $y$ increases with $x$. How to obtain such a function
Rewriting your question as a monotonic function from $[j,k]$ to $[0,1]$ with $f(j)=0$ and $f(k)=1$ where $j \lt k$, then $$f(x)= \dfrac{x-j}{k-j}$$ does this simply