Chapter 1.2, Problem 17E

(a)

To determine

To find:The domain and range of the function $y=x^{\frac{2}{5}}$ .

Answer to Problem 17E

The domain and range of the function $y=x^{\frac{2}{5}}$ is $\left(-\infty ,\infty \right)$ and $\left[0,\infty \right)$ respectively.

Explanation of Solution

Given information:The given function is $y=x^{\frac{2}{5}}$ .

Calculation:

Consider the given function.

  $y=x^{\frac{2}{5}}$

Domain is the set of all input values. The given function is defined for all real values of $x$ .So, the domain of the function is the set of real numbers $\left(-\infty ,\infty \right)$ .

Range is the set of all output values. The function $y$ is defined for positive input values. So, the range of the functionis $\left[0,\infty \right)$ .

Therefore, the domain and range of the function $y=x^{\frac{2}{5}}$ is $\left(-\infty ,\infty \right)$ and $\left[0,\infty \right)$ respectively.

(b)

To determine

To graph:The function $y=x^{\frac{2}{5}}$ with the help of graphing calculator.

Explanation of Solution

Given information:The function is $y=x^{\frac{2}{5}}$ .

Graph:

To graph a function $f\left(x\right)=x^{\frac{2}{5}}$ , 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 $f\left(x\right)=x^{\frac{2}{5}}$ 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 given quadratic equation in standard window as shown in Figure (1).

  

Figure (1)

Interpretation:The graph of the function $y=x^{\frac{2}{5}}$ is the curve with minimum value $0$ and it is positive for all real values of $x$ .