Chapter 7, Problem 38SGR

To determine

To Graph :

Graph the given inequality.

Explanation of Solution

Given:

The inequality is:

  $y\ge \sqrt{x}+3$

Concept Used:

• Definition of nth root:

• Words : For any real numbers a and b , and any positive integer n, if $a^n=b$ then a is the nth root of b. That is $a=\sqrt[n]{b}$
• $\begin{array}{l}If\text{ n is even: }\\ \sqrt[n]{b}\ge 0\text{ and }b\ge 0\end{array}$
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$
• For the inequality , if given y-value is greater ,then the shaded region will be above the boundary of the inequality.
• If given y-value is less , then the shaded region will be below the boundary of the inequality.

Graph:

  $\begin{array}{l}Given\\ y\ge \sqrt{x}+3\\ \text{Since the index is 2, which is even; so the radicand is non-negative}\text{.}\\ x\ge 0\\ Also\text{, the radical value will be non-negative }\text{.}\\ \sqrt{x}\ge 0\\ \sqrt{x}+3\ge \mathrm{3........}[\text{add 3 each side]}\\ y\ge \sqrt{x}+3\ge 3\\ So,\text{ for the inequality , }\\ \text{Domain= {x | x}\ge \text{0}}\\ \text{Range = { y | y }\ge \text{ 3}}\\ \text{First we draw the boundary }\\ y=\sqrt{x}+3\\ \text{The graph of }y=\sqrt{x}+3\text{ is the graph of }y=\sqrt{x}\\ shifted\text{ 3 units upward since the vertex is (0,3)}\text{.}\\ \text{Since the range is { y | y }\ge \text{ 3}},\\ the\text{ shaded region or the solution region will}\\ \text{be above the boundary line}\text{.}\end{array}$