Suppose that the functions $f$ and $g$ are defined as follows. \[ \begin{array}{l} f(x)=\sqrt{3 x+1} \\ g(x)=4 x^{2}+3 \end{array} \] Find $f \cdot g$ and $f+g$. Then, give their domains using interval notation.

Step 1 ：First, we find the product of the functions $f$ and $g$, denoted as $f \cdot g$. This is done by multiplying the expressions for $f(x)$ and $g(x)$ together.

Step 2 ：$f \cdot g = f(x) \cdot g(x) = \sqrt{3x+1} \cdot (4x^2+3)$

Step 3 ：Next, we find the sum of the functions $f$ and $g$, denoted as $f+g$. This is done by adding the expressions for $f(x)$ and $g(x)$ together.

Step 4 ：$f+g = f(x) + g(x) = \sqrt{3x+1} + 4x^2+3$

Step 5 ：Now, we find the domain of $f \cdot g$ and $f+g$. The domain of a function is the set of all possible input values (x-values) which will produce a valid output.

Step 6 ：The domain of $f(x)$ is $x \geq -1/3$, because the expression under the square root must be non-negative. The domain of $g(x)$ is all real numbers, because any real number can be squared and added to 3.

Step 7 ：Therefore, the domain of $f \cdot g$ and $f+g$ is the intersection of the domains of $f$ and $g$, which is $x \geq -1/3$ in interval notation, or $[-1/3, \infty)$.

Step 8 ：So, the functions $f \cdot g$ and $f+g$ are $f \cdot g = \sqrt{3x+1} \cdot (4x^2+3)$ and $f+g = \sqrt{3x+1} + 4x^2+3$, respectively, and their domain is $[-1/3, \infty)$.