Question

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

Answer 1

Domain: `(-oo, + oo)`

Explanation:

Since you're dealing with the square root of an expression, you know that you need to exclude from the domain of the function any value of `x` that will make the expression under the square root negative.

For real numbers, the square root can only be taken from positive numbers, which means that you need

`x^2 - 2x + 5 >=0`

Now you need to find the values of `x` for which the above inequality is satisfied. Look what happens when you use a little algebraic manipulation to rewrite the inequality

`x^2 - 2x + 5 >= 0`

`x^2 - 2x + 1 + 4 >=0`

`(x-1)^2 + 4 >=0`

Because `(x-1)^2 >=0` for any value of `x in RR`, it follows that

`(x-1)^2 + 4 >=0", "(AA)x in RR`

This means that the domain of the function can include all real numbers, since you cannot have a negative expression under the square root regardless of which `x` you plug in.

In interval notation, the domain of the function will thus be `(-oo, + oo)`.

graph{sqrt(x^2-2x+5) [-10, 10, -5, 5]}