1. **State the problem:** We have the function $f(x,y) = x^2 e^{3xy}$. (a) Evaluate $f(1,1)$. 2. **Evaluate $f(1,1)$:** Substitute $x=1$ and $y=1$ into the function: $$f(1,1) = 1^2 e^{3 \cdot 1 \cdot 1} = 1 \cdot e^3 = e^3$$ 3. **Find the domain of $f$:** The function involves $x^2$ and $e^{3xy}$. Since $x^2$ is defined for all real $x$ and $e^{3xy}$ is defined for all real $x,y$, the domain is all real pairs $(x,y)$: $$\text{Domain} = \{(x,y) \in \mathbb{R}^2\}$$ 4. **Find the range of $f$:** - Since $x^2 \geq 0$ for all $x$, and $e^{3xy} > 0$ for all real $x,y$, the function $f(x,y) = x^2 e^{3xy} \geq 0$. - When $x=0$, $f(0,y) = 0$ for all $y$. - For $x \neq 0$, $f(x,y) > 0$. - By choosing $y$ appropriately, $e^{3xy}$ can be made arbitrarily large or small (but positive), so $f$ can take any positive value. Therefore, the range is: $$\text{Range} = [0, \infty)$$