Solving the domain and range of a region satisfying two inequalities? The question I was provided was:
"Find the domain and range of the region satisfied by the following inequalities: 
i) $y \ge (x-1)^2$
ii)$y \le2x+1$
Any help would be greatly appreciated. Would you recommend graphing or solving algebraically?
 A: I am giving you a very basic way to find out the regions graphically. For $$y\ge(x-1)^2$$ Note that $y=(x-1)^2$ is a polynomial which has $\mathbb R$ as domain and obviously $\mathbb R_{\ge 0}$ as its range. To find out what that inequality tells you, you can pick two points in and out of the region the parabola made in $\mathbb R^2$. As you see $P$ is in and $Q$ is out with respect the parabola. Now satisfy the coordinates of $P$ into the inequality. You see $$(1/2-1)^2=1/4$$ and it is smaller than $2$. The same for $Q$ tells us $$(3-1)^2=4$$ is greater than $1$. So, the desired region satisfying $$y\ge(x-1)^2$$ is the parabola an the region which is enclosed by it.

A: 
I explain more detail:
first ,let $y=2x+1,y=(x-1)^2$, you will have two points$(0,1)$ and $(4,9)$
$ y \le 2x+1$ means all points below the red line, for example, $(0,0),(1,0),(1,1)$
$y \ge (x-1)^2$ means all points within the Parabola，for example $(1,1)(2,2)$
so two curves will have a closed area. the domain and range are: $ [0,4]$ and $[0,9]$
