Question

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

Answer 1

Domain: `x in RRcolor(white)("xxxxxx")or -oo < x < +oo`
Range: `f(x) in RRcolor(white)("xxx.x")or -oo < f(x) < +oo`

Explanation:

`ln e^color(red)a=color(red )a`

So `ln e^color(red)(2x)=color(red)(2x)`
and
`f(x)=ln e^(2x)`
`color(white)("xxx")rarr`f(x)=2x#

`f(x)` is defined for all Real values of `x` (establishing the Domain as all `RR`)

Any Real value, `2x`, can be established by for some Real value of `x` (establishing the Range as all `RR`)