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

Dec 15, 2017

Domain: $x \le - 2$ or $x \ge 2$ or $\left(- \infty , 2\right] \cup \left[2 , \infty\right)$
Range: All real numbers or $\left(- \infty , \infty\right)$

#### Explanation:

Domain refers to all the x-values on the graph.
In a square root function, to get real x-values, we need it to be $0$ ($\sqrt{0} = 0$ or any positive number. Therefore, we will set the square root stuff equal to or more than $0$, like this:
$\sqrt{{x}^{2} - 4} \ge 0$
${x}^{2} - 4 \ge 0$
${x}^{2} \ge 4$
$x \le - 2$ or $x \ge 2$ or $\left(- \infty , 2\right] \cup \left[2 , \infty\right)$

Range refers to all the y-values on the graph. Since all the values must be greater than or equal to 0, that means that the range is all real numbers, or $\left(- \infty , \infty\right)$

Here is the graph of this function (there should be arrows at the ends of the graph):