1. **Problem Statement:** Find the composite function $ (f \circ g)(x) $ where $ f(x) = \frac{x+1}{x-3} $ and $ g(x) = x + 3 $. Also, determine the domain of $ (f \circ g)(x) $. 2. **Formula and Rules:** The composite function $ (f \circ g)(x) $ means $ f(g(x)) $. This means we substitute $ g(x) $ into $ f(x) $ wherever $ x $ appears. 3. **Find $ (f \circ g)(x) $:** $$ (f \circ g)(x) = f(g(x)) = f(x+3) = \frac{(x+3)+1}{(x+3)-3} = \frac{x+4}{x} $$ 4. **Domain of $ (f \circ g)(x) $:** The domain of $ (f \circ g)(x) $ consists of all $ x $ values for which the expression is defined. - The denominator cannot be zero: $$ x \neq 0 $$ - Also, the domain of $ g(x) $ is all real numbers since $ g(x) = x+3 $ is defined everywhere. - The domain of $ f(x) $ is all real numbers except where the denominator is zero: $$ x - 3 \neq 0 \Rightarrow x \neq 3 $$ - For $ f(g(x)) $, the denominator is $ (x+3) - 3 = x $, so $ x \neq 0 $. 5. **Final answer:** $$ (f \circ g)(x) = \frac{x+4}{x}, \quad \text{with domain } x \in \mathbb{R} \setminus \{0\} $$