005 Sample Final A, Question 2

Question Find the domain of the following function. Your answer should be in interval notation $f(x)={\frac {1}{\sqrt {x^{2}-x-2}}}$

Foundations:
1) What is the domain of ${\frac {1}{\sqrt {x}}}$?
2) How can we factor $x^{2}-x-2$?
Answer:
1) The domain is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0,\infty )} . The domain of ${\frac {1}{x}}$ is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, \infty)} , but we have to remove zero from the domain since we cannot divide by 0.
2) $x^{2}-x-2=(x-2)(x+1)$

Step 1:
We start by factoring $x^{2}-x-2$ into $(x-2)(x+1)$

Step 2:
Since we cannot divide by zero, and we cannot take the square root of a negative number, we use a sign chart to determine when $(x-2)(x+1)>0$
$x:$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x<-1} $x=-1$ $-1<x<2$ $x=2$ $2<x$ $Sign:$ $(+)$ $0$ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (+)}

Step 3:
Now we just write, in interval notation, the intervals over which the denominator is positive.
The domain of the function is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1) \cup (2, \infty)}

Final Answer:
The domain of the function is: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, -1) \cup (2, \infty)}

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