Questions: Find the domain of the following function. g(x) = 6 / sqrt(15+x) The domain is □ (Type your answer in interval notation.)

Solution Steps

The function given is

\[ g(x) = \frac{6}{\sqrt{15 + x}} \]

To find the domain, we need to identify the values of \(x\) for which the function is defined. The function involves a square root in the denominator, which imposes two restrictions:

1. The expression inside the square root must be non-negative, i.e., \(15 + x \geq 0\).
2. The denominator cannot be zero, i.e., \(\sqrt{15 + x} \neq 0\).

First, solve the inequality \(15 + x \geq 0\):

\[ 15 + x \geq 0 \implies x \geq -15 \]

This inequality ensures that the expression inside the square root is non-negative.

Next, ensure that the denominator is not zero:

\[ \sqrt{15 + x} \neq 0 \implies 15 + x \neq 0 \]

Solving this gives:

\[ 15 + x \neq 0 \implies x \neq -15 \]

Combining the conditions from Steps 2 and 3, we find that:

• \(x \geq -15\) from the non-negativity condition.
• \(x \neq -15\) from the non-zero condition.

Thus, the domain of the function is all \(x\) such that \(x > -15\).

Final Answer