Question Find the domain of the following function. Your answer should be in interval notation $f(x)={\frac {1}{\sqrt {x^{2}x2}}}$
Foundations:

1) What is the domain of ${\frac {1}{\sqrt {x}}}$?

2) How can we factor $x^{2}x2$?

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) $x^{2}x2=(x2)(x+1)$

Step 1:

We start by factoring $x^{2}x2$ into $(x2)(x+1)$

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 $(x2)(x+1)>0$

$x:$ 
$x<1$ 
$x=1$ 
$1<x<2$ 
$x=2$ 
$2<x$ 
${\text{Sign: }}$ 
$(+)$ 
$0$ 
$()$ 
$0$ 
$(+)$ 

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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