Given functions \( g(x)=\frac{1}{\sqrt{x}} \) and \( p(x)=x^{2}-4 \), state the domains of the following functions using interval notation. Domain of \( \frac{g(x)}{p(x)}:(0,2) \cup(2, \infty) \) Domain of \( g(p(x)):(-\infty,-2) \cup(2, \infty) \) Domain of \( p(g(x)): \square \)

To determine the domain of \( p(g(x)) \), let's break down the functions and their compositions. ### Given Functions: - \( g(x) = \frac{1}{\sqrt{x}} \) - \( p(x) = x^2 - 4 \) ### Composition \( p(g(x)) \): \[ p(g(x)) = p\left(\frac{1}{\sqrt{x}}\right) = \left(\frac{1}{\sqrt{x}}\right)^2 - 4 = \frac{1}{x} - 4 \] ### Determining the Domain: 1. **Inner Function \( g(x) \):** - \( \sqrt{x} \) is defined for \( x > 0 \). - The expression \( \frac{1}{\sqrt{x}} \) requires \( x > 0 \) to avoid division by zero. 2. **Composite Function \( p(g(x)) \):** - Since \( p(x) \) is a polynomial, it's defined for all real numbers. - Therefore, the domain of \( p(g(x)) \) is solely determined by the domain of \( g(x) \). ### Final Domain: \[ \text{Domain of } p(g(x)): \ (0, \infty) \] **Interval Notation:** \( (0, \infty) \)