Chapter 2, Problem 62RE

To determine

To sketch: the graph of the function. Also state the domain, intercepts and range of the graph.

Answer to Problem 62RE

The domain of the function $f\left(x\right)$ is $\left\{x|x\ge -3\right\}$ .

The range is $\left\{y|y\le 0\right\}$ .

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=-\sqrt{x+3}$

Calculation:

Let us consider the following function

  $f\left(x\right)=-\sqrt{x+3}$

The function $f\left(x\right)=-\sqrt{x+3}$ has the same shape as the graph of $g\left(x\right)=\sqrt{x}$ .

The graph of $f\left(x\right)=-\sqrt{x+3}$ can be obtained from the graph of $g\left(x\right)=\sqrt{x}$ by applying the following steps.

1. Shift horizontally left 3 units.

Reflect about $x$

2. -axis.

The graph $f\left(x\right)=-\sqrt{x+3}$ is plotted below using Maple.

  

The given function is $f\left(x\right)=-\sqrt{x+3}$ .The function $f\left(x\right)=-\sqrt{x+3}$ tells us to take the square root of $x+3$ and then multiply by $-1$ . But only nonnegative numbers have real square roots, so the expression under the square root must be nonnegative. This requires that

  $\begin{array}{c}x+3\ge 0\\ x\ge -3\end{array}$

The domain of the function $f\left(x\right)$ is $\left\{x|x\ge -3\right\}$ .

We see $f\left(x\right)$ can take any values less than or equal to $0$ . Hence the range is $\left\{y|y\le 0\right\}$ .