# How do you find the domain and range of sqrt(x^2- 4)?

Dec 13, 2017

The domain: $\left(- \infty , - 2\right] \bigcup \left[2 , \infty\right)$
the range: $\left[0 , \infty\right)$

#### Explanation:

$f \left(x\right) = \sqrt{{x}^{2} - 4}$
The best and fastest way is to learn how do parental functions look like and how does the formula look like and then use it.
The domain: square root must be greater or equal to 0:

${x}^{2} - 4 \ge 0 \quad \implies \quad \left(x - 2\right) \left(x + 2\right) \ge 0$

That happens only when f(x)>0......everything that is above x axis, like this:(i usually draw a simple picture of quadratic function)
$x \in \left(- \infty , - 2\right] \bigcup \left[2 , \infty\right)$ (everything that is red)

There are many ways to find the range. I do it like this: parental square root starts in $\left({x}_{0} , {y}_{0}\right) \implies \left(0 , 0\right)$ and goes up and curving to the positive values, like this: graph{sqrt(x) [-10, 10, -5, 5]}
As you can see. It is always positive. so the range is: $\left[0 , \infty\right)$