Chapter 10, Problem 3CR

To determine

To graph the function and find Domain, Range.

Answer to Problem 3CR

Domain = $[-3,\infty )$

Range = $(-\infty ,-1]$

Explanation of Solution

Given information: $r\left(x\right)=-\sqrt{x+3}-1$

Formula Used: Square root function cannot be negative.

Calculation:

  $r\left(x\right)=-\sqrt{x+3}-1$

As the domain of square root function is greater than or equal to zero i.e. cannot be negative.

So,

  $\sqrt{x+3}\ge 0$

Squaring both sides,

  $x+3\ge 0$

Now,

  $x\ge -3$

Hence,

Domain = $[-3,\infty )$

As,

  $\sqrt{x+3}\ge 0$

So,

  $-\sqrt{x+3}\le 0$

Now, Subtract 1 both sides

  $-\sqrt{x+3}-1\le 0-1$

So,

  $-\sqrt{x+3}-1\le -1$

As, $r\left(x\right)=-\sqrt{x+3}-1$

  $r\left(x\right)\le -1$

Range = $(-\infty ,-1]$

To find the transformation, compare the function to the parent function and check to see if there is a horizontal or vertical shift, reflection about the x-axis or y-axis.

Parent function: $f\left(x\right)=\sqrt{x}$

Horizontal shift: left 3 units

Vertical shift: down 1 unit

Stretch or shrink: shrink.