# 005 Sample Final A, Question 2

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

Foundations:
1) What is the domain of ${\displaystyle {\frac {1}{\sqrt {x}}}}$?
2) How can we factor ${\displaystyle x^{2}-x-2}$?
1) The domain is ${\displaystyle (0,\infty )}$. The domain of ${\displaystyle {\frac {1}{x}}}$ is ${\displaystyle [0,\infty )}$, but we have to remove zero from the domain since we cannot divide by 0.
2) ${\displaystyle x^{2}-x-2=(x-2)(x+1)}$

Step 1:
We start by factoring ${\displaystyle x^{2}-x-2}$ into ${\displaystyle (x-2)(x+1)}$
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 ${\displaystyle (x-2)(x+1)>0}$
 ${\displaystyle x:}$ ${\displaystyle x<-1}$ ${\displaystyle x=-1}$ ${\displaystyle -1 ${\displaystyle x=2}$ ${\displaystyle 2 ${\displaystyle {\text{Sign: }}}$ ${\displaystyle (+)}$ ${\displaystyle 0}$ ${\displaystyle (-)}$ ${\displaystyle 0}$ ${\displaystyle (+)}$
Step 3:
Now we just write, in interval notation, the intervals over which the denominator is positive.
The domain of the function is: ${\displaystyle (-\infty ,-1)\cup (2,\infty )}$
The domain of the function is: ${\displaystyle (-\infty ,-1)\cup (2,\infty )}$