# How do you find the domain and range of f(x)= lne^(2x)?

Sep 26, 2017

Domain: $x \in \mathbb{R} \textcolor{w h i t e}{\text{xxxxxx}} \mathmr{and} - \infty < x < + \infty$
Range: $f \left(x\right) \in \mathbb{R} \textcolor{w h i t e}{\text{xxx.x}} \mathmr{and} - \infty < f \left(x\right) < + \infty$

#### Explanation:

$\ln {e}^{\textcolor{red}{a}} = \textcolor{red}{a}$

So $\ln {e}^{\textcolor{red}{2 x}} = \textcolor{red}{2 x}$
and
$f \left(x\right) = \ln {e}^{2 x}$
$\textcolor{w h i t e}{\text{xxx}} \rightarrow$f(x)=2x#

$f \left(x\right)$ is defined for all Real values of $x$ (establishing the Domain as all $\mathbb{R}$)

Any Real value, $2 x$, can be established by for some Real value of $x$ (establishing the Range as all $\mathbb{R}$)