Z-domain transfer function question
I assume this is homework, but transforming a difference equation to the $z$-domain is simple; just recall the time-shifting property of the transform.

$$
x[n] \Leftrightarrow X(z) \rightarrow x[n-k] \Leftrightarrow z^{-k}X(z)
$$

So then we have:

$$
y[n] = \frac{1}{2}x[n] + x[n-1]
$$

$$
Y(z) = \frac{1}{2}X(z) + z^{-1}X(z)
$$

The transfer function can be written as:

$$
H(z) = \frac{Y(z)}{X(z)} = \frac{1}{2} + z^{-1}
$$