1. **State the problem:** Find the domains of the functions $f(x) = \sqrt[3]{2x+6}$ and $g(x) = \sqrt[4]{2x+2}$. 2. **Recall domain rules for radicals:** - For odd roots (like cube roots), the radicand (expression inside the root) can be any real number. - For even roots (like fourth roots), the radicand must be greater than or equal to zero. 3. **Domain of $f(x) = \sqrt[3]{2x+6}$:** - Since it is a cube root (odd root), $2x+6$ can be any real number. - So, domain of $f$ is all real numbers. - In interval notation: $$(-\infty, \infty)$$ 4. **Domain of $g(x) = \sqrt[4]{2x+2}$:** - Since it is a fourth root (even root), the radicand must be $\geq 0$. - Set inequality: $$2x + 2 \geq 0$$ - Solve for $x$: $$2x \geq -2$$ $$x \geq -1$$ - So, domain of $g$ is all $x$ such that $x \geq -1$. - In interval notation: $$[-1, \infty)$$ **Final answers:** - Domain of $f$: $$(-\infty, \infty)$$ - Domain of $g$: $$[-1, \infty)$$