Question

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

Answer 1

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[`.