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Find the domain of the function $$f(x) = \frac{1}{x+8} + \frac{1}{\sqrt{x - 8}} + \frac{1}{\sqrt{8 - x}}.$$ Express your answer as a union of intervals.

 Oct 13, 2023
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The function f(x)=x+81​+x−8​1​+8−x​1​ is undefined when any of the denominators are zero. The first denominator, x+8, is zero when x=−8. The second denominator, x−8​, is zero when x=8. The third denominator, 8−x​, is zero when x=8. Therefore, the function is undefined when x=−8 or x=8.

We also need to consider the restrictions imposed by the square root functions. The square root of a number is undefined when the number is negative. Therefore, the second denominator, x−8​, is undefined when x−8<0, which is when x<8. Similarly, the third denominator, 8−x​, is undefined when 8−x<0, which is when x>8.

Putting all of this together, the function f(x) is undefined when x=−8, x=8, x<8, or x>8. Therefore, the domain of f(x) is the union of the intervals (−∞,−8)∪(8,∞)​.

 Oct 13, 2023

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