Answer
Linear function.
Work Step by Step
$g(x,y,z) = \frac{xz+3yz+z^2}{4z}$
As $z\ne 0$ we can divide the numerator by $4z$. This gives us:
$g(x,y,z)=\frac{x+3y+z}{4}$
None of the terms has the product of two variables in it and all of them are linear, therefore it is a linear function.
