1. We are asked to determine if the function $$g(x) = \frac{7x^2}{10 + 2x}$$ is continuous on various intervals. 2. Recall that a rational function is continuous everywhere its denominator is not zero. 3. Find where the denominator is zero: $$10 + 2x = 0 \implies 2x = -10 \implies x = -5$$ 4. So, $$g(x)$$ is continuous everywhere except at $$x = -5$$. 5. Now check each interval: - a) $$(-6, 6)$$ includes $$-5$$, so $$g(x)$$ is not continuous on $$(-6, 6)$$. - b) $$[-10, 0]$$ includes $$-5$$, so $$g(x)$$ is not continuous on $$[-10, 0]$$. - c) $$(-\infty, -5)$$ does not include $$-5$$ (endpoint not included), so $$g(x)$$ is continuous on $$(-\infty, -5)$$. - d) $$[-10, 20)$$ includes $$-5$$, so $$g(x)$$ is not continuous on $$[-10, 20)$$. 6. Therefore, $$x \neq -5$$ for continuity. --- 7. Next, determine if $$m(x) = \sqrt{4 - x}$$ is continuous on given intervals. 8. The square root function is continuous on its domain, which requires the radicand to be non-negative: $$4 - x \geq 0 \implies x \leq 4$$ 9. So, $$m(x)$$ is continuous for all $$x \leq 4$$. 10. Check each interval: - a) $$(0, 5)$$ includes values greater than 4, so $$m(x)$$ is not continuous on $$(0, 5)$$. - b) $$[-6, 2]$$ is entirely less than or equal to 4, so $$m(x)$$ is continuous on $$[-6, 2]$$. - c) $$[3, 9)$$ includes values greater than 4, so $$m(x)$$ is not continuous on $$[3, 9)$$. - d) $$(-\infty, 4)$$ is within the domain, so $$m(x)$$ is continuous on $$(-\infty, 4)$$. 11. Therefore, $$x \neq 4$$ for continuity. Final answers: - For $$g(x)$$, discontinuous at $$x = -5$$. - For $$m(x)$$, discontinuous at $$x = 4$$.