Chapter 1.2, Problem 19E

(a)

To determine

To find:The domain and range of the function $y=\sqrt[3]{x-3}$ .

Answer to Problem 19E

The domain and range of the function $y=\sqrt[3]{x-3}$ is $\left(-\infty ,\infty \right)$ and $\left(-\infty ,\infty \right)$ respectively.

Explanation of Solution

Given information:The given function is $y=\sqrt[3]{x-3}$ .

Calculation:

Consider the given function.

  $y=\sqrt[3]{x-3}$

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

Range is the set of all output values. The function $y$ is always a real number for all input values. So, the range of the function is the set of real numbers $\left(-\infty ,\infty \right)$ .

Therefore, the domain and range of the function $y=\sqrt[3]{x-3}$ is $\left(-\infty ,\infty \right)$ and $\left(-\infty ,\infty \right)$ respectively.

(b)

To determine

To graph:The function $y=\sqrt[3]{x-3}$ with the help of graphing calculator.

Explanation of Solution

Given information:The function is $y=\sqrt[3]{x-3}$ .

Graph:

To graph a function $f\left(x\right)=\sqrt[3]{x-3}$ , 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 $f\left(x\right)=\sqrt[3]{x-3}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{3}\boxed{)}\boxed{^}\boxed{(}\boxed{1}\boxed{÷}\boxed{3}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=\left(\text{X}-\text{3}\right)^\left(1/3\right)$

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 below:

  

Figure (1)

Interpretation:The graph of the function $y=\sqrt[3]{x-3}$ is similar to the graph of function $\sqrt[3]{x}$ with $x$ -intercept as $3$ and $y$ -intercept as $-1$ .