To determine the domain of the function \(\frac{f}{g}(x)\) where \(f(x) = \frac{1}{x}\) and \(g(x) = \sqrt{x + 8}\), we need to consider the constraints from both \(f(x)\) and \(g(x)\).
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For \(f(x) = \frac{1}{x}\):
- The function \(f(x)\) is defined for all \(x\) except \(x = 0\).
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For \(g(x) = \sqrt{x + 8}\):
- The square root function is defined when the expression inside the square root is non-negative. Therefore, we need: \[ x + 8 \geq 0 \implies x \geq -8. \]
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Combining the constraints:
- From \(g(x)\), \(x\) must be greater than or equal to \(-8\).
- From \(f(x)\), \(x\) cannot be equal to \(0\).
Thus, the domain of \(\frac{f}{g}(x)\) is:
- \(x \in [-8, 0) \cup (0, \infty)\).
Therefore, the answer is: \[ \text{left parenthesis } -8, 0 \text{ right parenthesis union left parenthesis 0, infinity \text{ right parenthesis.} \]
So the final answer is: left parenthesis negative 8 comma 0 right parenthesis union left parenthesis 0 comma infinity right parenthesis.