If $Y$ is a function of $X$, is it possible the two variables are independent?

Question

If I have a variable $X$ and a variable $Y$ where $Y = \alpha + \beta + X$ and $\alpha$ and $\beta$ are constants is there any scenario in which $X$ and $Y$ are independent?

Intuitively, I feel the answer is no, meaning Cov($X$,$Y$) $\ne$ $0$, but I want to be sure.