Chapter 2 - Functions, Equations, and Graphs - 2-6 Families of Functions - Practice and Problem-Solving Exercises - Page 104: 23

$g(x)=-x-1$

Recall: (1) The graph of $y=f(-x)$ involves a reflection about the $y$-axis of the parent function $f(x)$. (2) The graph of $y=-f(x)$ involves a reflection about the $x$-axis of the parent function $f(x)$. Thus, when $f(x)=x+1$ is reflected about the $x$-axis, the equation of the new function $g(x)$, is: \begin{align*} g(x)&=-f(x)\\ g(x)&=-(x+1)\\ g(x)&=-x-1 \end{align*}