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

Answer 1

Domain: #(-oo,oo)#

Range: #(-oo,2)#

The domain is all possible values of #x# with which #f(x)# is defined.
Here, any value of #x# will result in a defined function. Therefore, the domain is #-oo<##x<##oo#, or, in interval notation:
#(-oo,oo)#.
The range is all possible values of #f(x)#. It can also be defined as the domain of #f^-1(x)#.
So to find #f^-1(x):#
#y=2-e^(x/2)#
Interchange the variables #x# and #y#:
#x=2-e^(y/2)#
And solve for #y#:
#x-2=-e^(y/2)#
#e^(y/2)=2-x#

Take the natural logarithm of both sides:

#ln(e^(y/2))=ln(2-x)#
#y/2ln(e)=ln(2-x)#
As #ln(e)=1#,
#y/2=ln(2-x)#
#y=2ln(2-x)=f^-1(x)#

We must find the domain of the above.

For any #lnx,# #x>0#.
So here, #2-x>0#
#-x> -2#
#x##<##2#
So the range of #f(x)# can be stated as #(-oo,2)#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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