# How do you find the domain and range of g(x) = 1/(In x)?

Jul 3, 2017

Domain: $\left(0 , 1\right) \cup \left(1 , + \infty\right)$
Range: $\left(- \infty , + \infty\right)$

#### Explanation:

$g \left(x\right) = \frac{1}{\ln} x$

$\ln x$ is defined for $x > 0$

$\ln x = 0$ for $x = 1$ $\to g \left(x\right)$ is not defined for $x = 1$

Hence, $g \left(x\right)$ is defined for $x > 0 , x \ne 1$

$\therefore$ the domain of $g \left(x\right)$ is $\left(0 , 1\right) \cup \left(1 , + \infty\right)$

Now consider:

${\lim}_{\text{(x->1) -}} g \left(x\right) = - \infty$

${\lim}_{\text{(x->1) +}} g \left(x\right) = + \infty$

Hence, the range of $g \left(x\right)$ is $\left(- \infty , + \infty\right)$

We can see these results fron the graph of $g \left(x\right)$
below.

graph{1/lnx [-10, 10, -5.04, 4.96]}