1. **Stating the problem:** We are given the function $f(x) = \sqrt[3]{x - 8}$ and want to understand its properties. 2. **Formula and rules:** The cube root function $\sqrt[3]{x}$ is defined for all real numbers $x$ and is the inverse of the cube function $x^3$. 3. **Rewrite the function:** We can write $f(x)$ as $$f(x) = (x - 8)^{\frac{1}{3}}$$ 4. **Domain:** Since cube roots are defined for all real numbers, the domain of $f$ is all real numbers, $(-\infty, \infty)$. 5. **Intercepts:** - To find the $x$-intercept, set $f(x) = 0$: $$0 = (x - 8)^{\frac{1}{3}} \implies x - 8 = 0 \implies x = 8$$ So the $x$-intercept is at $(8, 0)$. - To find the $y$-intercept, set $x = 0$: $$f(0) = (0 - 8)^{\frac{1}{3}} = (-8)^{\frac{1}{3}} = -2$$ So the $y$-intercept is at $(0, -2)$. 6. **Behavior and shape:** The cube root function is increasing and continuous everywhere. The graph shifts 8 units to the right compared to $y = \sqrt[3]{x}$. 7. **Summary:** - Domain: all real numbers - $x$-intercept: $(8, 0)$ - $y$-intercept: $(0, -2)$ - The function is increasing and smooth. **Final answer:** The function $f(x) = \sqrt[3]{x - 8}$ has domain $(-\infty, \infty)$, $x$-intercept at $(8, 0)$, and $y$-intercept at $(0, -2)$.