Given function:

$f(x) = \sqrt{\frac{3x}{x+1}}$

Step 1: Identify Domain Restrictions

1. The function contains a square root, which means the expression inside must be non-negative: $\frac{3x}{x+1} \geq 0$
2. Solve the inequality:

   • Find where $\frac{3x}{x+1} = 0$:
     $3x = 0 \Rightarrow x = 0$

   • Find where $\frac{3x}{x+1}$ is undefined:
     $x + 1 = 0 \Rightarrow x = -1$

   • Determine sign changes by analyzing critical points $x = 0$ and $x = -1$:

     Interval	$3x$ Sign	$x+1$ Sign	$\frac{3x}{x+1}$ Sign
     $(-\infty, -1)$	$-$	$-$	$+$
     $(-1,0)$	$-$	$+$	$-$ (invalid)
     $(0,\infty)$	$+$	$+$	$+$

   • The function is valid for: $(-\infty, -1) \cup [0, \infty)$

Step 2: Check for Continuity

• The function is undefined at $x = -1$, so it is not continuous there.
• The function is undefined for $-1 < x < 0$ because the expression inside the square root is negative.
• For $x \in (-\infty, -1) \cup [0, \infty)$, the function is well-defined and continuous.

Final Answers:

Part 1: Where is the function continuous or not continuous?

• The function is not continuous for all values of $x$ between $-1$ and $0$ because the expression inside the square root is negative.
• The correct choice is:
  F. The function is not continuous for all values of $x$ less than 0 and greater than or equal to -1.

Part 2: Why is the function not continuous?

• The function is not continuous because it is not defined for $x$ in the interval $-1 \leq x < 0$.
• The correct choice is:
  A. The function is not continuous because it is not defined for $x$-values on the interval $-1 \leq x < 0$.

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Follow-up Questions for Better Understanding

1. How do you determine when a function is undefined?
2. What are the general conditions for continuity at a point?
3. How does the square root function affect domain restrictions?
4. What is the difference between continuity and differentiability?
5. How can you use limits to formally prove continuity?

Tip: Always check for restrictions in denominators and square roots when determining function domains.