Question: Find the domains of the functions \( f \) and \( g \). \[ f(x) = \sqrt[4]{x - 4} \] \[…

Find the domains of the functions \( f \) and \( g \).

\[ f(x) = \sqrt[4]{x - 4} \] \[ g(x) = \sqrt[3]{3x + 3} \]

Write your answers using interval notation.

Domain of \( f \): \(\boxed{ \quad } \)

Domain of \( g \): \(\boxed{ \quad } \)

Solution

To find the domains of the functions \( f(x) \) and \( g(x) \), consider the following: For \( f(x) = \sqrt[4]{x - 4} \): The expression under the fourth root must be non-negative, because the fourth root is only defined for non-negative numbers. Set up the inequality: \[ x - 4 \geq 0 \] Solving for \( x \): \[ x \geq 4 \] Thus, the domain of \( f(x) \) is \([4, \infty)\). For \( g(x) = \sqrt[3]{3x + 3} \): The cube root is defined for all real numbers, so there is no restriction for \( 3x + 3 \). Thus, the domain of \( g(x) \) is \((- \infty, \infty)\). Final answers: Domain of \( f \): \([4, \infty)\) Domain of \( g \): \((- \infty, \infty)\)