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

We are given   g(x) = 1/√x  and  m(x) = x² – 4. When finding the domain of a function constructed from these, we must ensure every expression involved is defined. (Remember: the square root √u is defined only for u ≥ 0, and if it appears in the denominator, we must have u > 0. Also, division by zero is never allowed.) Let’s analyze each function in turn. ────────────────────────────── 1. Domain of g(x)/m(x) Here, the function is   [g(x)]/[m(x)] = (1/√x) / (x² – 4). (a) For g(x) = 1/√x to be defined, we need √x to exist and not be 0:   • x must be ≥ 0 for √x to exist;   • x must be > 0 because if x = 0 then √0 = 0 and 1/0 is undefined. Thus, x > 0. (b) For the quotient, the denominator m(x) = x² – 4 must not be 0:   x² – 4 ≠ 0 ⟹ x² ≠ 4 ⟹ x ≠ 2 and x ≠ –2. However, since we already require x > 0, x = –2 is automatically excluded. So we must exclude x = 2. Therefore, the domain of g(x)/m(x) is all x > 0 except x = 2. In interval notation, that is   (0, 2) ∪ (2, ∞). ────────────────────────────── 2. Domain of g(m(x)) Here, we compose the functions: g(m(x)) = 1/√(m(x)) = 1/√(x² – 4). For g(u) = 1/√u to be defined, we need the argument u to be strictly positive (u > 0). Here, u = m(x) = x² – 4. So we require   x² – 4 > 0. Solve the inequality:   x² > 4 ⟹ |x| > 2. That is, x < –2 or x > 2. So in interval notation, the domain is   (–∞, –2) ∪ (2, ∞). ────────────────────────────── 3. Domain of m(g(x)) Now, m(g(x)) = m(1/√x) = (1/√x)² – 4. First, find when g(x) = 1/√x is defined. As before, we need   x > 0. Next, m is a polynomial defined for all real numbers. Therefore, no additional restrictions come from m. So the domain is simply the domain of g(x), which is   (0, ∞). ────────────────────────────── Summary of Answers: 1. Domain of g(x)/m(x):  (0, 2) ∪ (2, ∞) 2. Domain of g(m(x)):  (–∞, –2) ∪ (2, ∞) 3. Domain of m(g(x)):  (0, ∞) Each answer is now given using correct interval notation.