1. **State the problem:** We are given the function $$f(x) = \log_{4} \left( x^{2} - 1 \right)$$ and want to understand its properties and graph shape. 2. **Domain determination:** The logarithm function is defined only for positive arguments. So, we require: $$x^{2} - 1 > 0$$ This inequality can be rewritten as: $$x^{2} > 1$$ which means: $$x < -1 \quad \text{or} \quad x > 1$$ 3. **Behavior near domain boundaries:** At $$x = \pm 1$$, the argument of the logarithm is zero, so the function has vertical asymptotes there. 4. **Shape of the graph:** Since $$x^{2} - 1$$ grows large as $$|x|$$ increases, the logarithm will increase slowly for large $$|x|$$. 5. **Summary:** The function has two branches: - For $$x < -1$$, $$f(x)$$ is defined and decreases towards $$-\infty$$ as $$x \to -1^{-}$$ and increases slowly as $$x \to -\infty$$. - For $$x > 1$$, $$f(x)$$ is defined and increases slowly as $$x \to \infty$$ and decreases towards $$-\infty$$ as $$x \to 1^{+}$$. This matches the described graph with vertical asymptotes at $$x = \pm 1$$ and two branches on either side. Final answer: The domain is $$(-\infty, -1) \cup (1, \infty)$$ and the graph has vertical asymptotes at $$x = -1$$ and $$x = 1$$ with the described logarithmic shape on each branch.