Chapter 4 - Inverse, Exponential, and Logarithmic Functions - Summary Exercises on Functions: Domains and Defining Equations - Exercises - Page 487: 26

$(-\infty,-\sqrt 7)\cup(-\sqrt 7,\sqrt 7)\cup(\sqrt 7,\infty)$.

The domain requirement for $f(x)=\frac{1}{|x^2-7|}$ is $|x^2-7|\ne0$ or $x\ne\pm\sqrt 7$ , in interval notation $(-\infty,-\sqrt 7)\cup(-\sqrt 7,\sqrt 7)\cup(\sqrt 7,\infty)$.