1. **Problem statement:** We are given three functions: - $f(x) = x^4 + x + 2$ - $g(x) = \frac{1}{3x + 1}$ - $h(x) = \frac{1}{\sqrt[3]{x - 2}}$ We need to find the domains of $f$, $g$, and $h$, then find the intersections of these domains: - $\text{Dom}_f \cap \text{Dom}_g$ - $\text{Dom}_f \cap \text{Dom}_h$ - $\text{Dom}_g \cap \text{Dom}_h$ 2. **Finding $\text{Dom}_f$:** The function $f(x) = x^4 + x + 2$ is a polynomial, and polynomials are defined for all real numbers. Therefore, $$\text{Dom}_f = \mathbb{R}$$ 3. **Finding $\text{Dom}_g$:** The function $g(x) = \frac{1}{3x + 1}$ is defined for all real numbers except where the denominator is zero. Set denominator equal to zero: $$3x + 1 = 0$$ Solve for $x$: $$3x = -1$$ $$x = \cancel{\frac{-1}{3}}$$ So $x = -\frac{1}{3}$ is excluded. Therefore, $$\text{Dom}_g = \mathbb{R} \setminus \left\{-\frac{1}{3}\right\}$$ 4. **Finding $\text{Dom}_h$:** The function $h(x) = \frac{1}{\sqrt[3]{x - 2}}$ involves the cube root in the denominator. Cube root $\sqrt[3]{x - 2}$ is defined for all real $x$, but since it is in the denominator, it cannot be zero. Set denominator equal to zero: $$\sqrt[3]{x - 2} = 0$$ Cube both sides: $$x - 2 = 0$$ $$x = \cancel{2}$$ So $x = 2$ is excluded. Therefore, $$\text{Dom}_h = \mathbb{R} \setminus \{2\}$$ 5. **Find $\text{Dom}_f \cap \text{Dom}_g$:** Since $\text{Dom}_f = \mathbb{R}$ and $\text{Dom}_g = \mathbb{R} \setminus \left\{-\frac{1}{3}\right\}$, $$\text{Dom}_f \cap \text{Dom}_g = \mathbb{R} \setminus \left\{-\frac{1}{3}\right\}$$ 6. **Find $\text{Dom}_f \cap \text{Dom}_h$:** Since $\text{Dom}_f = \mathbb{R}$ and $\text{Dom}_h = \mathbb{R} \setminus \{2\}$, $$\text{Dom}_f \cap \text{Dom}_h = \mathbb{R} \setminus \{2\}$$ 7. **Find $\text{Dom}_g \cap \text{Dom}_h$:** $$\text{Dom}_g = \mathbb{R} \setminus \left\{-\frac{1}{3}\right\}$$ $$\text{Dom}_h = \mathbb{R} \setminus \{2\}$$ Their intersection excludes both points: $$\text{Dom}_g \cap \text{Dom}_h = \mathbb{R} \setminus \left\{-\frac{1}{3}, 2\right\}$$ **Final answers:** - $\text{Dom}_f = \mathbb{R}$ - $\text{Dom}_g = \mathbb{R} \setminus \left\{-\frac{1}{3}\right\}$ - $\text{Dom}_h = \mathbb{R} \setminus \{2\}$ - $\text{Dom}_f \cap \text{Dom}_g = \mathbb{R} \setminus \left\{-\frac{1}{3}\right\}$ - $\text{Dom}_f \cap \text{Dom}_h = \mathbb{R} \setminus \{2\}$ - $\text{Dom}_g \cap \text{Dom}_h = \mathbb{R} \setminus \left\{-\frac{1}{3}, 2\right\}$