What is the domain and range of #f(x) =sqrt (x^2 - 2x + 5)#?

1 Answer
Nov 10, 2015

Answer:

Domain : #RR#.
Range : #[2,+oo[#.

Explanation:

The domain of #f# is the set of real #x# such that #x^2-2x+5>=0#.

You write #x^2-2x+5 = (x-1)^2 +4# (canonical form), so you can see that #x^2-2x+5 >0# for all real #x#. Therefore, the domain of #f# is #RR#.

The range is the set of all values of #f#. Because #x mapsto sqrt(x)# is an increasing function, the variations of #f# are same than #x mapsto (x-1)^2+4# :
- #f# is increasing on #[1,+oo[#,
- #f# is decreasing on #]-oo,1]#.
The minimal value of #f# is #f(1) = sqrt(4)=2#, and f has no maximum.

Finally, the range of #f# is #[2,+oo[#.