Determining the Domain of 4/√(x+8): A Function Analysis

To solve this problem, we need to determine the domain of the function given by:

$f(x) = \frac{4}{\sqrt{x+8}}$

We must ensure that the function is defined for all $x$. This involves considering the conditions under which the square root is valid and the denominator is non-zero.

Step 1: Analyze the square root expression $\sqrt{x+8}$. The expression inside the square root must be non-negative:

$x + 8 \ge 0$

Step 2: Solve the inequality:

• Subtract 8 from both sides: $x \ge -8$

Step 3: Consider the division by zero issue. The denominator $\sqrt{x+8}$ must be strictly greater than zero to avoid division by zero. Thus, we adjust the inequality to:

$x + 8 > 0$

Step 4: Solve the second inequality:

• Subtract 8 from both sides: $x > -8$

Thus, the domain of the function is all $x$ such that $x > -8$.

Review of the answer choices shows that the correct choice, consistent with our findings, is:

$x > -8$

Therefore, the domain of the function is $x > -8$.