Chapter 0.2, Problem 10E

(a)

To determine

To find: The domain and range of the given function.

Answer to Problem 10E

The domain is equal to $\left(-\infty ,0\right]$ and the range is equal to $\left[0,\infty \right)$ .

Explanation of Solution

Given information:

The function is $y=\sqrt[4]{-x}$ .

Calculation:

The given function is $y=\sqrt[4]{-x}$ .

To find the domain: There does not exist a real the root of a negative number, so the inside of the square root, $-x$ , must be greater than or equal to zero.

It can be simplified as below:

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

So, the $y$ exists for all values of $x$ which are less than or equal to $0$ .

Therefore, the domain is all real number less than or equal to $0$ .

  $x\in \left(-\infty ,0\right]$

To find the range: The range of the function will be determined by the fact that the square root of a real number must always be positive.

The smallest value the function can take will happen for $x=0$ , since those values of $x$ will make the radical term equal to zero.

  $y=\sqrt[4]{-0}=0$ .

The range of the function will be $\left[0,\infty \right)$ .

Hence, the domain is $\left(-\infty ,0\right]$ and the range is $\left[0,\infty \right)$ .

(b)

To determine

To sketch: The graph of the function $y=\sqrt[4]{-x}$ .

Explanation of Solution

Given information:

The function is $y=\sqrt[4]{-x}$ .

Calculation:

The given function is $y=\sqrt[4]{-x}$ .

The given function is the parabolic function with vertex $\left(0,0\right)$ .

Now, sketch the graph of the function below: