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: Understanding the Concept:
A function is continuous on an interval if it is defined and has no breaks, jumps, or holes in that interval. For a function involving square roots, the primary concern for continuity is the domain. The function will be continuous wherever it is defined. We need to find the domain of \( f(x) \).
Step 2: Key Formula or Approach:
The domain of a square root function \( g(x) = \sqrt{h(x)} \) is the set of all x for which \( h(x) \geq 0 \). When a function is a combination of multiple functions (like a product or sum), its domain is the intersection of the domains of the individual functions.
Step 3: Detailed Explanation:
The function is \( f(x) = x(\sqrt{x+2} + \sqrt{x+1}) \). It is composed of three parts: \( g(x)=x \), \( h(x)=\sqrt{x+2} \), and \( k(x)=\sqrt{x+1} \).
1. The domain of \( g(x)=x \) is all real numbers, \( (-\infty, \infty) \).
2. For \( h(x)=\sqrt{x+2} \) to be defined, the expression inside the root must be non-negative:
\[ x+2 \geq 0 \implies x \geq -2 \] The domain is \( [-2, \infty) \).
3. For \( k(x)=\sqrt{x+1} \) to be defined, the expression inside the root must be non-negative:
\[ x+1 \geq 0 \implies x \geq -1 \] The domain is \( [-1, \infty) \).
The domain of the entire function \( f(x) \) is the intersection of the domains of all its parts. We need to find the values of x that satisfy all three conditions simultaneously.
- \( x \in (-\infty, \infty) \)
- \( x \in [-2, \infty) \)
- \( x \in [-1, \infty) \)
The intersection of these three intervals is the most restrictive one, which is \( [-1, \infty) \).
The function is composed of standard continuous functions (polynomials and square roots) combined through multiplication and addition. Therefore, it is continuous throughout its domain.
Step 4: Final Answer:
The function is continuous on the interval \( [-1, \infty) \).