005 Sample Final A, Question 2

Question Find the domain of the following function. Your answer should be in interval notation 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 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 $(0,\infty )$. The domain of ${\frac {1}{x}}$ is $[0,\infty )$, 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 $(x-2)(x+1)>0$
$x:$ $x<-1$ $x=-1$ $-1<x<2$ $x=2$ $2<x$ ${\text{Sign: }}$ $(+)$ $0$ $(-)$ $0$ $(+)$

Step 3:
Now we just write, in interval notation, the intervals over which the denominator is positive.
The domain of the function is: $(-\infty ,-1)\cup (2,\infty )$

Final Answer:
The domain of the function is: $(-\infty ,-1)\cup (2,\infty )$

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