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 $(0,\infty )$. 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 $(x-2)(x+1)>0$
$x:$ $x<-1$ $x=-1$ $-1<x<2$ $x=2$ $2<x$ $Sign:$ $(+)$ $0$ $(-)$ $0$ $(+)$

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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