\begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{ Finding the Domain of Radical Functions } \\ \hline \multicolumn{3}{|c|}{\begin{tabular}{l} Determine the domain for each of the following functions. Write your answer in Interval Notation and as an \\ Inequality. \end{tabular}} \\ \hline Radical Function & \begin{tabular}{l} Domain written in Interval \\ Notation \end{tabular} & \begin{tabular}{l} Domain written as an \\ Inequality \end{tabular} \\ \hline$f(x)=\sqrt{20 x-15}$ & & \\ \hline$g(x)=8 \sqrt{-15 x+18}$ & & \\ \hline$f(x)=-4+\sqrt{3 x+1}$ & & \\ \hline$p(x)=\sqrt{-14+2 x}$ & & \\ \hline \end{tabular}

Step 1 ：Given the functions $f(x)=\sqrt{20 x-15}$, $g(x)=8 \sqrt{-15 x+18}$, $f(x)=-4+\sqrt{3 x+1}$, and $p(x)=\sqrt{-14+2 x}$, we need to find the domain for each of these functions. The domain of a radical function is the set of all real numbers for which the function is defined. For a square root function, the expression under the square root (the radicand) must be greater than or equal to zero, because the square root of a negative number is not a real number.

Step 2 ：To find the domain of each function, we need to solve the inequality $radicand \geq 0$. For the function $f(x)=\sqrt{20 x-15}$, the radicand is $20x - 15$. Solving the inequality $20x - 15 \geq 0$ gives us $x \geq 3/4$. So, the domain of $f(x)$ in interval notation is $[3/4, \infty)$ and as an inequality is $20x - 15 \geq 0$.

Step 3 ：For the function $g(x)=8 \sqrt{-15 x+18}$, the radicand is $-15x + 18$. Solving the inequality $-15x + 18 \geq 0$ gives us $x \leq 6/5$. So, the domain of $g(x)$ in interval notation is $(-\infty, 6/5]$ and as an inequality is $18 - 15x \geq 0$.

Step 4 ：For the function $f(x)=-4+\sqrt{3 x+1}$, the radicand is $3x + 1$. Solving the inequality $3x + 1 \geq 0$ gives us $x \geq -1/3$. So, the domain of $f(x)$ in interval notation is $[-1/3, \infty)$ and as an inequality is $3x + 1 \geq 0$.

Step 5 ：For the function $p(x)=\sqrt{-14+2 x}$, the radicand is $2x - 14$. Solving the inequality $2x - 14 \geq 0$ gives us $x \geq 7$. So, the domain of $p(x)$ in interval notation is $[7, \infty)$ and as an inequality is $2x - 14 \geq 0$.

Step 6 ：Final Answer: For $f(x)=\sqrt{20 x-15}$, the domain in interval notation is \(\boxed{[3/4, \infty)}\) and as an inequality is \(\boxed{20x - 15 \geq 0}\). For $g(x)=8 \sqrt{-15 x+18}$, the domain in interval notation is \(\boxed{(-\infty, 6/5]}\) and as an inequality is \(\boxed{18 - 15x \geq 0}\). For $f(x)=-4+\sqrt{3 x+1}$, the domain in interval notation is \(\boxed{[-1/3, \infty)}\) and as an inequality is \(\boxed{3x + 1 \geq 0}\). For $p(x)=\sqrt{-14+2 x}$, the domain in interval notation is \(\boxed{[7, \infty)}\) and as an inequality is \(\boxed{2x - 14 \geq 0}\).