Question

The function $f(x)=x(\sqrt{x+2}+\sqrt{x+1})$ is continuous on

• A. $(-\infty,1]$
• B. $[4,\infty)$
• C. $[-3,\infty)$
• D. $[-1,\infty)$
• E. $(-\infty,\infty)$

Hint:

The domain of $f+g$ is the intersection of the domains of $f$ and $g$.

Answer 1

The Correct Option is D

Step 1: Concept
A function involving square roots is continuous where the radicands are non-negative.

Step 2: Analysis
For $\sqrt{x+2}$ to be defined: $x+2 \ge 0 \implies x \ge -2$. For $\sqrt{x+1}$ to be defined: $x+1 \ge 0 \implies x \ge -1$.

Step 3: Conclusion
The domain is the intersection of these conditions: $x \ge -1$. Thus, the function is continuous on $[-1, \infty)$. Final Answer: (D)