Chapter 7.2, Problem 51PPS

(a)

To determine

To Graph:

  $y=x^n\text{ for n=0,1,2,3,4}$ using a graphing calculator.

Explanation of Solution

Graph:

  $y=x^0\Rightarrow y=1$

  

  $y=x^1\Rightarrow y=x$

  

  $y=x^2$

  

  $y=x^3$

  

  $y=x^4$

  

(b)

To determine

To Find:

The values of n for which the inverse is a function.

Explanation of Solution

Given Information:

From part (a) , we have the graphs of each function for the values n = 0,1,2,3,4

Concept Used:

The horizontal line test states that if the graph of a function $y=f(x)$ is such that no horizontal line intersects the graph in more than one point, then the function has an inverse function , or the inverse of the function $y=f(x)$ is a function.

Calculation:

For each graph , we use the horizontal line test to check if the inverse is a function or not.

• n=0
$y=x^0\Rightarrow y=1$

From the graph , the line y = 1 intersects the graph at more than one point.So, by the horizontal line test , the inverse is not a function.

• n = 1
$y=x^1\Rightarrow y=x$

From the graph , there is no horizontal line that intersects the graph at more than one point. So, by the horizontal line test , the inverse is a function.

• n = 2
$y=x^2$

From the graph , the line y = 2 intersects the graph at more than one point. So, by the horizontal line test , the inverse is not a function.

• n = 3
$y=x^3$

From the graph , there is no horizontal line that intersects the graph at more than one point. So, by the horizontal line test , the inverse is a function.

• n = 4
$y=x^4$

From the graph , the line y = 2 intersects the graph at more than one point. So, by the horizontal line test , the inverse is not a function.

Summarizing the above results in the table :

Function	Inverse is a function?
$y=x^0\Rightarrow y=1$	No
$y=x^1\Rightarrow y=x$	Yes
$y=x^2$	No
$y=x^3$	Yes
$y=x^4$	No

(c)

To determine

To Make:

A conjecture about the values of n for which the inverse of $f(x)=x^n$ is a function. Assume that n is a whole number.

Answer to Problem 51PPS

When n is odd , the inverse of the function is a function.

Explanation of Solution

Given Information:

From part (b) , we have

$y=x^0\Rightarrow y=1$	No
$y=x^1\Rightarrow y=x$	Yes
$y=x^2$	No
$y=x^3$	Yes
$y=x^4$	No

Calculation:

So, clearly , from the table , we have that when n is odd, the inverse is a function , since it passes the horizontal line test.

Hence, when n is odd, the inverse of the function is a function.