1. The problem is to analyze the function $f(x) = \frac{10}{7x^3 - 2}$ where $x \in \mathbb{R}$. 2. This is a rational function where the numerator is a constant 10 and the denominator is a cubic polynomial $7x^3 - 2$. 3. Important rules: - The function is undefined where the denominator is zero. - To find domain, solve $7x^3 - 2 = 0$. 4. Solve for $x$: $$7x^3 - 2 = 0 \implies 7x^3 = 2 \implies x^3 = \frac{2}{7} \implies x = \sqrt[3]{\frac{2}{7}}$$ 5. So the function is undefined at $x = \sqrt[3]{\frac{2}{7}}$. 6. The domain is all real numbers except $x = \sqrt[3]{\frac{2}{7}}$. 7. To find intercepts: - $y$-intercept: set $x=0$, then $f(0) = \frac{10}{7\cdot0 - 2} = \frac{10}{-2} = -5$. - $x$-intercept: set $f(x) = 0$, numerator must be zero, but numerator is 10, so no $x$-intercepts. 8. Summary: - Domain: $\mathbb{R} \setminus \left\{ \sqrt[3]{\frac{2}{7}} \right\}$ - $y$-intercept: $(0, -5)$ - No $x$-intercepts - Vertical asymptote at $x = \sqrt[3]{\frac{2}{7}}$ Final answer: $$f(x) = \frac{10}{7x^3 - 2}, \quad x \in \mathbb{R} \setminus \left\{ \sqrt[3]{\frac{2}{7}} \right\}$$