1. **State the problem:** Analyze the key features of the function $$g(x) = \sqrt[3]{x - 8} + 4$$ and determine which statements A to E are true. 2. **Recall the parent function:** The parent cube root function is $$f(x) = \sqrt[3]{x}$$. - Domain: all real numbers - Range: all real numbers - No absolute minimum or maximum - Graph passes through origin (0,0) 3. **Analyze transformations:** - Inside the cube root, $$x - 8$$ means a horizontal shift **right** by 8 units. - The $$+4$$ outside means a vertical shift **up** by 4 units. 4. **Domain of $$g(x)$$:** - Cube root functions are defined for all real numbers. - Since $$x - 8$$ is inside the cube root, domain is all real numbers. - So, domain is $$(-\infty, \infty)$$, not $$x \geq 8$$. 5. **Range of $$g(x)$$:** - Cube root functions have range all real numbers. - Vertical shift up by 4 does not change range. - So, range is all real numbers. 6. **Absolute minimum:** - Cube root functions have no absolute minimum or maximum. - So, no absolute minimum at $$x=8$$. 7. **Behavior as $$x \to \infty$$:** - As $$x \to \infty$$, $$x - 8 \to \infty$$, so $$\sqrt[3]{x - 8} \to \infty$$. - Adding 4 shifts the output up but does not change the limit. - So, $$g(x) \to \infty$$ as $$x \to \infty$$. 8. **Summary of statements:** - A: True (range is all real numbers) - B: False (no absolute minimum) - C: False (domain is all real numbers) - D: False (graph is shifted right 8 units, not left) - E: True (as $$x \to \infty$$, $$g(x) \to \infty$$) **Final answers:** A and E are true.