Chapter 10, Problem 4CT

To determine

To graph the function and find Domain, Range.

Answer to Problem 4CT

Domain = $[-6,\infty )$

Range = $(-\infty ,0]$

Explanation of Solution

Given information: $f\left(x\right)=-\sqrt{x+6};g\left(x\right)=\sqrt{x}$

Formula Used: Square root function cannot be negative.

Calculation:

  $f\left(x\right)=-\sqrt{x+6};g\left(x\right)=\sqrt{x}$

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

So,

  $\sqrt{x+6}\ge 0$

Now, squaring both sides,

  $x+6\ge 0$

So,

  $x\ge -6$

Hence,

Domain = $[-6,\infty )$

As,

  $\sqrt{x+6}\ge 0$

Now,

  $-\sqrt{x+6}\le 0$

As, $f\left(x\right)=-\sqrt{x+6}$

  $f\left(x\right)\le 0$

Range = $(-\infty ,0]$

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: $g\left(x\right)=\sqrt{x}$

Horizontal shift: left 6 units

Vertical shift: none

Stretch or shrink: shrink.