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What are all possible values of #f(x) = 1/x^2-2x+3#?

What are all possible values of #f(x) = 1/x^2-2x+3#?
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ReportAnswer #1

#(-oo, +oo)#

Explanation:

#f(x) = 1/x^2-2x+3#

We are asked to find "all possible values" of #f(x)#

Assuming #f(x) in RR# this means the range of #f(x): x in RR#

First let's find the domain of #f(x)#

#f(x)# is defined #forall x in RR: x!=0#

#:. # the domain of #f(x)# is #(-oo,0)uu(0,+oo)#

Now consider,

#lim_(x->0-) f(x) = +oo# and. #lim_(x->0+) f(x) =+oo#

Also,

#lim_(x->-oo) f(x) = +oo#

and

#lim_(x->+oo) f(x) =-oo#

Hence the range of #f(x) = (-oo,+oo)#

This maybe more clearly seen from the graph of #f(x)# below.

graph{ 1/x^2-2x+3 [-28.88, 28.83, -14.43, 14.43]}

answered by: Alan N.
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