Chapter 10, Problem 6CT

To determine

To graph the function and find Domain, Range.

Answer to Problem 6CT

Domain = $[0,\infty )$

Range = $[-5,\infty )$

Explanation of Solution

Given information: $f\left(x\right)=\sqrt[3]{x}-5;g\left(x\right)=\sqrt[3]{x}$

Formula Used: Square root function cannot be negative.

Calculation:

  $f\left(x\right)=\sqrt[3]{x}-5;g\left(x\right)=\sqrt[3]{x}$

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

So,

  $\sqrt[3]{x}\ge 0$

Now, cubing both sides,

  $x\ge 0$

Hence,

Domain = $[0,\infty )$

As,

  $\sqrt[3]{x}\ge 0$

Now, Subtract 5 both sides

  $\sqrt[3]{x}-5\ge -5$

As, $f\left(x\right)=\sqrt[3]{x}-5$

  $f\left(x\right)\ge -5$

Range = $[-5,\infty )$

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: none

Vertical shift: down 5 units

Stretch or shrink: none.