# Technicality of describing where sqrt(x) is increasing

1. Feb 19, 2008

When describing where $sqrt{x}$ (square root of x) is increasing, it's from zero to infinity. But, do you say (0,inf) or [0, inf) ?

(I'm tutoring a student in pre-calc, and this came up. They don't know any calculus.)

In a situation like where is $x^2$ inc/dec, we'd say inc: (0, inf) and dec (-inf, 0). We wouldn't include the zero because that's the point where the function switches from dec to inc.

But on square root of x, it's not "switching" from dec to inc, so can we include that zero?

Thanks so much.

2. Feb 19, 2008

### D H

Staff Emeritus
A function f is monotonically increasing over some set S if for any $x<y \Rightarrow f(x) < f(y) \, \forall x,y \in S$. No need for derivatives here! With this definition, you could say [tex]\sqrt x[/itex] is a monitonically increasing function over the set $S=[0,\infty)$.

3. Feb 19, 2008

### Hurkyl

Staff Emeritus
And also, note that $x^2$ is strictly increasing on $[0, +\infty)$!