Domain of a quadratic function under square root having no x intercept

[ziaharipur]

I am trying to find the domain of this function f(x)=sqrt(x^2 - 2x +5). i am supposing that y=x^2 - 2x +5 and y must be greater of equal to 0. in otherwords where the graph is touching or above x axis. to find where the graph is touching x-axis i am trying to find the x intercept and i get complex x intercept, does this mean that graph is not touching x axis? now if this happens how can we find out that the graph is above x-axis or below x-axis please explain...

 

[Tedjn]

Saying y(x) has no x-intercept is the same thing as saying that y(x) is never equal to 0. The quadratic y(x) is continuous; if you have studied or are studying calculus, you will know that from the intermediate value theorem that y(x) is therefore always positive or always negative. If you have not studied calculus yet, you may still know that the graph of y(x) is a parabola; no real roots means the parabola is either always above or below the x-axis. The domain of f(x) just requires y(x) > 0. In this case, how do you confirm when y(x) > 0? When in doubt, also graph y(x) to assure yourself.