Chapter 1.4, Problem 29E

a.

To determine

Graph the function.

Answer to Problem 29E

  

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\sqrt[n]{x}$ , where $n$ is a positive integer. Graph the function $y=\sqrt{x},y=\sqrt[4]{x},and,y=\sqrt[6]{x}$ on the same screen using the viewing rectangle $\left[-1,4\right]$ by $\left[-1,3\right]$

Calculation:

Consider the given function, $f\left(x\right)=\sqrt[n]{x}$ and sketch the graph for $y=\sqrt{x},y=\sqrt[4]{x},and,y=\sqrt[6]{x}$ . using the viewing rectangle $\left[-1,4\right]$ by $\left[-1,3\right]$ .

Use $TI-83$ calculator,

Press $\boxed{Y=}$ button and enter the equation,

  

Press on $\boxed{WINDOW}$ button and set the viewing rectangle as shown below,

  

Press $\boxed{GRAPH}$ button,

  

Hence, the result shown in the graph.

b.

To determine

Graph the function.

Answer to Problem 29E

  

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\sqrt[n]{x}$ , where $n$ is a positive integer. Graph the function $y=x,y=\sqrt[3]{x},and,y=\sqrt[5]{x}$ on the same screen using the viewing rectangle $\left[-3,3\right]$ by $\left[-2,2\right]$

Calculation:

Consider the given function, $f\left(x\right)=\sqrt[n]{x}$ and sketch the graph for $y=x,y=\sqrt[3]{x},and,y=\sqrt[5]{x}$ . using the viewing rectangle $\left[-3,3\right]$ by $\left[-2,2\right]$ .

Use $TI-83$ calculator,

Press $\boxed{Y=}$ button and enter the equation,

  

Press on $\boxed{WINDOW}$ button and set the viewing rectangle as shown below,

  

Press $\boxed{GRAPH}$ button,

  

Hence, the result shown in the graph.

c.

To determine

Graph the function $y=\sqrt{x},y=\sqrt[3]{x},y=\sqrt[4]{x},and,y=\sqrt[5]{x}$ on the same screen using the viewing rectangle $\left[-1,3\right]$ by $\left[-1,2\right]$ .

Answer to Problem 29E

  

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\sqrt[n]{x}$ , where $n$ is a positive integer. Graph the function $y=\sqrt{x},y=\sqrt[3]{x},y=\sqrt[4]{x},and,y=\sqrt[5]{x}$ on the same screen using the viewing rectangle $\left[-1,3\right]$ by $\left[-1,2\right]$ .

Calculation:

Consider the given function, $f\left(x\right)=\sqrt[n]{x}$ and sketch the graph for $y=\sqrt{x},y=\sqrt[3]{x},y=\sqrt[4]{x},and,y=\sqrt[5]{x}$ . Using the viewing rectangle $\left[-1,3\right]$ by $\left[-1,2\right]$ .

Use $TI-83$ calculator,

Press $\boxed{Y=}$ button and enter the equation,

  

Press on $\boxed{WINDOW}$ button and set the viewing rectangle as shown below,

  

Press $\boxed{GRAPH}$ button,

  

Hence, the result shown in the graph.

d.

To determine

What conclusion can you make from these graphs?

Answer to Problem 29E

As $n$ increases, the graph of $y=\sqrt[n]{x}$ become steeper near zero and flatter for $x>1$ .

Explanation of Solution

Given information:

We consider the family of root function $f\left(x\right)=\sqrt[n]{x}$ , where $n$ is a positive integer. What conclusion can you make from these graphs?

Calculation:

It is observed from the all graphs that even root graphs are similar and odd root graphs are similar.

Hence,As $n$ increases, the graph of $y=\sqrt[n]{x}$ become steeper near zero and flatter for $x>1$ .