Determine the interval(s) for which the function is continuous.

• A. $[-3,0),[0,\infty )$
• B. $(-∞,0),[0,∞)$
• C. $(-∞,∞)$
• D. The function is not continuous anywhere.

Answer: (A)

Verified step by step guidance

1. Identify the piecewise function: f(x) = sqrt(9 - x^2) for -3 <= x < 0 and f(x) = 5 for x >= 0.
2. Determine the continuity of the first piece: sqrt(9 - x^2) is continuous for -3 <= x < 0 because it is a composition of continuous functions (square root and polynomial).
3. Determine the continuity of the second piece: The constant function f(x) = 5 is continuous for all x >= 0.
4. Check the continuity at the boundary x = 0: The left-hand limit as x approaches 0 from the left is sqrt(9 - 0^2) = 3, and the right-hand limit as x approaches 0 from the right is 5. Since these limits are not equal, f(x) is not continuous at x = 0.
5. Combine the intervals of continuity: The function is continuous on the intervals [-3, 0) and [0, ∞).