Chapter 1, Problem 34RE

(a)

To determine

To find: The domain of the function $y=x^{2/5}$ .

Answer to Problem 34RE

The domain of the function $y=x^{2/5}$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=x^{2/5}$ .

Calculation:

The function $y=x^{2/5}$ is defined for all the real values of $x$ .

Therefore, the domain of the function $y=x^{2/5}$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(b)

To determine

To find: The range of the function $y=x^{2/5}$ .

Answer to Problem 34RE

The range of the function $y=x^{2/5}$ is the interval $\left[0,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=x^{2/5}$ .

Calculation:

The function $f\left(x\right)=x^2$ is always greater than or equal to $0$ .

Raise power $\frac{1}{5}$ both sides.

  $\begin{array}{c}{\left(f\left(x\right)\right)}^{\frac{1}{5}}={\left(x^2\right)}^{\frac{1}{5}}\\ =x^{\frac{2}{5}}\\ =y\end{array}$

So, the function $y=x^{\frac{2}{5}}$ is also greater than or equal to $0$ .

Therefore, the range of the function $y=x^{2/5}$ is the interval $\left[0,\infty \right)$ .

(c)

To determine

To graph: The function $y=x^{2/5}$ .

Explanation of Solution

Given information:

The given functionis $y=x^{2/5}$ .

Graph:

To graph a function $y=x^{2/5}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=x^{2/5}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{^}\boxed{(}\boxed{2}\boxed{÷}\boxed{5}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\text{X}^\left(2/5\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph the function $y=x^{2/5}$ is symmetric about the y -axis.