Z-transform of an affine function What is the transfer function of the system described by the following affine input ($x$)-output ($y$) relationship:
$$
y[n] = \alpha x[n] + \beta.
$$
Using the Z-transform we find:
$$
Y[z] = \alpha X[z]+ \beta
$$
and
$$
H[z] = \frac{Y[z]}{X[z]} = \alpha + \frac{\beta}{X[z]}
$$
But I don't know how to take it from here to compute the transfer function! I'm not sure if this can be done. Any help would be appreciated.
That system cannot be described by a transfer function because it is not linear. It neither satisfies the condition of additivity nor does it satisfy the condition of homogeneity.
If $y_1[n]$ is the response to an input $x_1[n]$, and $y_2[n]$ is the response to an input $x_2[n]$, then the response to $x_1[n]+x_2[n]$ doesn't equal $y_1[n]+y_2[n]$ (additivity is violated). Furthermore, the response to $\alpha x_1[n]$ doesn't equal $\alpha y_1[n]$ (homogeneity is violated).
This is unrelated to existence questions concerning the $\mathcal{Z}$-transform. The $\mathcal{Z}$-transform of a constant $\beta$ doesn't equal $\beta$, but it simply doesn't exist.