1. **State the problem:** We are given two functions $f(x) = \frac{x+4}{x}$ and $g(x) = x+3$. We need to find the composition $(f \circ g)(x)$, which means $f(g(x))$, and then determine its domain. 2. **Recall the composition formula:** The composition $(f \circ g)(x)$ is defined as $f(g(x))$. This means we substitute $g(x)$ into $f$. 3. **Calculate $f(g(x))$:** $$ (f \circ g)(x) = f(g(x)) = f(x+3) = \frac{(x+3)+4}{x+3} = \frac{x+7}{x+3} $$ 4. **Determine the domain:** - The domain of $g(x) = x+3$ is all real numbers. - The domain of $f(x) = \frac{x+4}{x}$ is all real numbers except $x \neq 0$ because division by zero is undefined. - For $f(g(x))$, the input to $f$ is $g(x) = x+3$. So we must have $x+3 \neq 0$ to avoid division by zero. 5. **Domain of $(f \circ g)(x)$:** $$ x+3 \neq 0 \implies x \neq -3 $$ 6. **Final answer:** $$(f \circ g)(x) = \frac{x+7}{x+3}, \quad \text{with domain } \{x \in \mathbb{R} : x \neq -3\}$$