What is the domain and range for $f(x) = 2 - e ^ (x / 2)$ ?

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Answer 1 · 2015-07-08T20:19:27

$f(x) : RR -> ]-oo;2[$

$f(x) = 2 - e^(x/2)$

Domain : $e^x$ is defined on $RR$ .
And $e^(x/2) = e^(x*1/2) = (e^(x))^(1/2) = sqrt(e^x)$ then $e^(x/2)$ is defined on $RR$ too.

And so, the domain of $f(x)$ is $RR$

Range :

The range of $e^x$ is $RR^(+)-{0}$ .

Then :

$0< e^x < +oo$
$<=> sqrt(0) < sqrt(e^x) < +oo$
$<=> 0 < e^(x/2) < +oo$
$<=> 0 > -e^(x/2) > -oo$
$<=> 2 > 2 -e^(x/2) > -oo$

Therefore,
$<=> 2 > f(x) > -oo$

What is the domain and range for f(x) = 2 - e ^ (x / 2)f(x) = 2 - e ^ (x / 2)?