# What is the domain for the function f(x)=1/(sqrtx-2)?

Jul 25, 2017

Domain: $\left[0 , 4\right) \cup \left(4 , + \infty\right)$
Range:: $\left(- \infty , - 0.5\right] \cup \left(0 , + \infty\right)$

#### Explanation:

$f \left(x\right) = \frac{1}{\sqrt{x} - 2}$

Considerations for the domain of $f \left(x\right)$

$\sqrt{x}$ is defined $\in \mathbb{R} \forall x \ge 0 \to$ Domain of $f \left(x\right) \ge 0$

$f \left(x\right)$ is undefined at $\sqrt{x} = 2 \to x \ne 4$

Combining these results:
the domain of $f \left(x\right) = \left[0 , 4\right) \cup \left(4 , + \infty\right)$

Considerations for the range of $f \left(x\right)$

$f \left(0\right) = - 0.5$
Since $x \ge 0 \to - 0.5$ is a local maximum of $f \left(x\right)$

${\lim}_{x \to {4}^{-}} f \left(x\right) = - \infty$

${\lim}_{x \to {4}^{+}} f \left(x\right) = + \infty$

${\lim}_{x \to + \infty} f \left(x\right) = 0$

Combining these results:
the range of $f \left(x\right) = \left(- \infty , - 0.5\right] \cup \left(0 , + \infty\right)$

These results can be observed by the graph of $f \left(x\right)$ below.

graph{1/(sqrtx-2) [-14.24, 14.24, -7.12, 7.12]}