Chapter 10, Problem 7CT

To determine

To graph the function and find Domain, Range.

Answer to Problem 7CT

Domain = $[-1,\infty )$

Range = $(-\infty ,0]$

Explanation of Solution

Given information: $f\left(x\right)=-2\sqrt[3]{x+1};g\left(x\right)=\sqrt[3]{x}$

Formula Used: Square root function cannot be negative.

Calculation:

  $f\left(x\right)=-2\sqrt[3]{x+1};g\left(x\right)=\sqrt[3]{x}$

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

So,

  $\sqrt[3]{x+1}\ge 0$

Now, cubing both sides,

  $x+1\ge 0$

So,

  $x\ge -1$

Hence,

Domain = $[-1,\infty )$

As,

  $\sqrt[3]{x+1}\ge 0$

Now,

  $-\sqrt[3]{x+1}\le 0$

So, multiply by 2 both sides

  $-2\sqrt[3]{x+1}\le 0$

As, $f\left(x\right)=-2\sqrt[3]{x+1}$

  $f\left(x\right)\le 0$

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: $g\left(x\right)=\sqrt[3]{x}$

Horizontal shift: left 1 unit

Vertical shift: none

Stretch or shrink: stretched.