Domain Range 381F3A

1. **State the problem:** Find the domain and range of the function $$f(x) = \sqrt{5x - 10}$$ and express them in set notation. 2. **Domain:** The expression inside the square root must be greater than or equal to zero because the square root of a negative number is not a real number. 3. Set the inequality: $$5x - 10 \geq 0$$ 4. Solve for $$x$$: $$5x \geq 10$$ $$\cancel{5}x \geq \cancel{10}2$$ $$x \geq 2$$ 5. **Domain in set notation:** $$\{ x \mid x \geq 2, x \in \mathbb{R} \}$$ 6. **Range:** Since $$f(x)$$ is a square root function, its output is always greater than or equal to zero. 7. The smallest value inside the root is zero (when $$x=2$$), so the smallest output is: $$f(2) = \sqrt{5(2) - 10} = \sqrt{10 - 10} = 0$$ 8. As $$x$$ increases, $$5x - 10$$ increases, so $$f(x)$$ increases without bound. 9. **Range in set notation:** $$\{ y \mid y \geq 0, y \in \mathbb{R} \}$$