Find the domains of the functions $f$ and $g$. \[ \begin{array}{l} f(x)=\sqrt[4]{2 x-6} \\ g(x)=\sqrt[3]{x-9} \end{array} \] Write your answers using interval notation.
Solution
Step 1 ：The domain of a function is the set of all possible input values (often the 'x' variable), which produce a valid output from a particular function. For the given functions, we need to find the values of x for which the functions are defined.
Step 2 ：For the function \(f(x)=\sqrt[4]{2 x-6}\), the expression under the fourth root must be greater than or equal to zero because we cannot take even roots of negative numbers. So, we need to solve the inequality \(2x - 6 \geq 0\).
Step 3 ：For the function \(g(x)=\sqrt[3]{x-9}\), the expression under the cube root can be any real number because we can take cube roots of negative numbers. So, the domain of this function is all real numbers.
Step 4 ：The solution to the inequality \(2x - 6 \geq 0\) is \(x \geq 3\). So, the domain of the function \(f(x)=\sqrt[4]{2 x-6}\) is \([3, \infty)\).
Step 5 ：For the function \(g(x)=\sqrt[3]{x-9}\), as we discussed earlier, the domain is all real numbers, which is \((-\infty, \infty)\).
Step 6 ：Final Answer: The domain of the function \(f(x)=\sqrt[4]{2 x-6}\) is \(\boxed{[3, \infty)}\) and the domain of the function \(g(x)=\sqrt[3]{x-9}\) is \(\boxed{(-\infty, \infty)}\).
