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#