How do you find domain and range for #f(x) =sqrt (x^2 - 2x + 5)#?

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Answer 1 · 2015-10-04T16:48:14

Complete the square to find that the domain of #f(x)# is the whole of #RR# and its range is #[2, oo)#

#f(x) = sqrt(x^2-2x+5) = sqrt((x-1)^2+4)#

#(x-1)^2+4 >= 4 > 0# for all #x in RR#

So #f(x) = sqrt((x-1)^2+4)# is defined for all #x in RR#

So the (implicit) domain of #f(x)# is #RR#.

#f(x)# has minimum value when #(x-1) = 0#, that is when #x = 1#.

#f(1) = sqrt(0^2+4) = sqrt(4) = 2#

So the range of #f(x)# is #[2, oo)#