How to Bound this Lipschitz Continuous Functions?

Question

Suppose $f:\mathbb{R} \rightarrow [0,1]$ is a Lipschitz Continuous Function with parameter $L$. Now, we define two variables $x\in [\underline{x},\bar{x}]$ and $y$ and $z=(x,y)$ where $y \in \mathbb{R}^n$ and $\|y\|_2 \le S$.The function $g$ is defined as $g(z)=xf(z^\top \gamma)$ where $\gamma$ is a known vector of parameters. I am wondering if $g$ will be a Lipschitz Continuous Function, and what its coefficient is. How is it possible to write inequality related to the Lipschitz constant? Thanks