Does the equation $y=1-e^{-kx}+xe^{-k}$ have a name?

Question

Today I've built this equation to solve an issue at work:

$y=1-e^{-kx}+xe^{-k}$

There's a demo at Desmos here!

It has three properties that I needed:

1. Always crosses $(0, 0)$ and $(1, 1)$
2. When $k$ is $0$ the function is a simple linear $y=x$.
3. When $k$ is greater than $0$ then the function grows quickly at the beginning and stops at $1$

Does this equation have a name? Are there a family of equations like this?

Answer 1

There is a similar family, $$ \big(1-(1-x)^k\big)^{1/k} $$ for $k=1$ it is a line segment,
for $k>1$ it grows rapidly at first, and ends at $1$
for $k=2$ it is an arc of a circle
for $0<k<1$ it grows slowly at first, and ends at $1$.

Pictures shows $k=\frac12,1,2,4$ (from bottom to top).

For more, see Superellipse