For \( f(x)=\frac{3}{x+7} \) and \( g(x)=\frac{2}{x} \), find a. \( (f \circ g)(x) ; \quad \) b. the domain of \( f \circ g \)

To find \( (f \circ g)(x) \), we substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f\left(\frac{2}{x}\right) = \frac{3}{\frac{2}{x} + 7} = \frac{3x}{2 + 7x}. \] Now for the domain of \( f \circ g \): To find the domain, we need to consider the values of \( x \) that keep \( g(x) \) and \( f(g(x)) \) defined. The function \( g(x) \) is undefined at \( x = 0 \), and \( f(x) \) is undefined for \( x + 7 = 0 \) (i.e., \( x = -7 \)). For \( f(g(x)) \), we need \( \frac{2}{x} + 7 \neq 0 \), leading to \( x \neq -\frac{2}{7} \). Therefore, the domain of \( f \circ g \) is \( x \in \mathbb{R} \setminus \{0, -\frac{2}{7}\} \).