1. The problem asks to find the composition function $fg(x)$ given two functions: - $f(x) = 3x + \ln x$, where $x > 0$ - $g(x) = e^{x^2}$, where $x \in \mathbb{R}$ 2. The composition $fg(x)$ means $f(g(x))$, which is applying $f$ to the output of $g(x)$. 3. Substitute $g(x)$ into $f$: $$fg(x) = f(g(x)) = f\left(e^{x^2}\right)$$ 4. Using the definition of $f$, replace $x$ by $e^{x^2}$: $$fg(x) = 3 \cdot e^{x^2} + \ln\left(e^{x^2}\right)$$ 5. Simplify the logarithm using the property $\ln\left(e^a\right) = a$: $$fg(x) = 3e^{x^2} + x^2$$ 6. Therefore, the composition function is: $$\boxed{fg(x) = x^2 + 3e^{x^2}}$$ This matches the given expression, confirming the result.