Chapter 2.2, Problem 65E

Solution

a.

To find: Plot the graph and the analyze the various features of the graph.

The graph of the functions is shown below.

  

The analysis of the graph is as follows:

	$f$	$g$	$h$	$k$
Domain	$x\ne 0$	$x\ne 0$	$x\ne 0$	$x\ne 0$
Range	$y\ne 0$	$y>0$	$y\ne 0$	$y>0$
Continuous	Yes	Yes	Yes	Yes
Increasing		$\left(-\infty ,0\right)$		$\left(-\infty ,0\right)$
Decreasing	$\left(-\infty ,0\right)$,$\left(0,\infty \right)$	$\left(-\infty ,0\right)$	$\left(-\infty ,0\right)$,$\left(0,\infty \right)$	$\left(-\infty ,0\right)$
Symmetry	Origin	Y-axis	Origin	Y-axis
Bounded	Not bounded	Below	Not bounded	Below
Extrema	No	No	No	No
Asymptotes	$x$-axis, $y$-axis	$x$-axis, $y$-axis	$x$-axis, $y$-axis	$x$-axis, $y$-axis
End behavior	$\underset{x\to \infty }{\lim }f\left(x\right)$	$\underset{x\to \infty }{\lim }g\left(x\right)$	$\underset{x\to \infty }{\lim }h\left(x\right)$	$\underset{x\to \infty }{\lim }k\left(x\right)$

Given:

The functions are $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ and the windows are $\left[0,1\right]$ by $\left[0,5\right]$ , $\left[0,3\right]$ by $\left[0,3\right]$ and $\left[-2,2\right]$ by $\left[-2,2\right]$ .

Calculation:

First of all plot all the functions $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ in the windows $\left[0,1\right]$ by $\left[0,5\right]$ .

Press the $Y=$ button and enter the functions $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ .

  

Set up the graphing window.

  

Press the $2^{nd}$ button and press $Y=$ button for the scatter plot of the data.

  

Next plot all the functions $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ in the windows $\left[0,3\right]$ by $\left[0,3\right]$ .

Press the $Y=$ button and enter the functions $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ .

  

Set up the graphing window.

  

Press the $2^{nd}$ button and press $Y=$ button for the scatter plot of the data.

  

Next plot all the functions $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ in the windows $\left[-2,2\right]$ by $\left[-2,2\right]$ .

Press the $Y=$ button and enter the functions $f\left(x\right)=x^{-1}$ , $g\left(x\right)=x^{-2}$ , $h\left(x\right)=x^{-3}$ , $k\left(x\right)=x^{-4}$ .

  

Set up the graphing window.

  

Press the $2^{nd}$ button and press $Y=$ button for the scatter plot of the data.

  

The analysis of the graphs is shown as below.

	$f$	$g$	$h$	$k$
Domain	$x\ne 0$	$x\ne 0$	$x\ne 0$	$x\ne 0$
Range	$y\ne 0$	$y>0$	$y\ne 0$	$y>0$
Continuous	Yes	Yes	Yes	Yes
Increasing		$\left(-\infty ,0\right)$		$\left(-\infty ,0\right)$
Decreasing	$\left(-\infty ,0\right)$ , $\left(0,\infty \right)$	$\left(-\infty ,0\right)$	$\left(-\infty ,0\right)$ , $\left(0,\infty \right)$	$\left(-\infty ,0\right)$
Symmetry	Origin	Y-axis	Origin	Y-axis
Bounded	Not bounded	Below	Not bounded	Below
Extrema	No	No	No	No
Asymptotes	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis
End behavior	$\underset{x\to \infty }{\lim }f\left(x\right)$	$\underset{x\to \infty }{\lim }g\left(x\right)$	$\underset{x\to \infty }{\lim }h\left(x\right)$	$\underset{x\to \infty }{\lim }k\left(x\right)$

b.

To find: Plot the graph and the analyze the various features of the graph.

The graph of the functions is shown below.

  

The analysis of the graph is as follows:

	$f$	$g$	$h$	$k$
Domain	$x\ne 0$	$x\ne 0$	$x\ne 0$	$x\ne 0$
Range	$y\ne 0$	$y>0$	$y\ne 0$	$y>0$
Continuous	Yes	Yes	Yes	Yes
Increasing		$\left(-\infty ,0\right)$		$\left(-\infty ,0\right)$
Decreasing	$\left(-\infty ,0\right)$, $\left(0,\infty \right)$	$\left(-\infty ,0\right)$	$\left(-\infty ,0\right)$, $\left(0,\infty \right)$	$\left(-\infty ,0\right)$
Symmetry	Origin	Y-axis	Origin	Y-axis
Bounded	Not bounded	Below	Not bounded	Below
Extrema	No	No	No	No
Asymptotes	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis
End behavior	$\underset{x\to \infty }{\lim }f\left(x\right)$	$\underset{x\to \infty }{\lim }g\left(x\right)$	$\underset{x\to \infty }{\lim }h\left(x\right)$	$\underset{x\to \infty }{\lim }k\left(x\right)$

Given:

The functions are $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$ and the windows are $\left[0,1\right]$ by $\left[0,1\right]$ , $\left[0,3\right]$ by $\left[0,2\right]$ and $\left[-3,3\right]$ by $\left[-2,2\right]$ .

Calculation:

First of all plot all the functions $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$ in the windows $\left[0,1\right]$ by $\left[0,1\right]$ .

Press the $Y=$ button and enter the functions $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$ .

  

Set up the graphing window.

  

Press the $2^{nd}$ button and press $Y=$ button for the scatter plot of the data.

  

Next plot all the functions $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$ in the windows $\left[0,3\right]$ by $\left[0,2\right]$ .

Press the $Y=$ button and enter the functions $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$ .

  

Set up the graphing window.

  

Press the $2^{nd}$ button and press $Y=$ button for the scatter plot of the data.

  

Next plot all the functions $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$ in the windows $\left[-3,3\right]$ by $\left[-2,2\right]$ .

Press the $Y=$ button and enter the functions $f\left(x\right)=x^{1/2}$ , $g\left(x\right)=x^{1/3}$ , $h\left(x\right)=x^{1/4}$ , $k\left(x\right)=x^{1/5}$

  

Set up the graphing window.

  

Press the $2^{nd}$ button and press $Y=$ button for the scatter plot of the data.

  

The analysis of the graphs is shown as below.

	$f$	$g$	$h$	$k$
Domain	$x\ne 0$	$x\ne 0$	$x\ne 0$	$x\ne 0$
Range	$y\ne 0$	$y>0$	$y\ne 0$	$y>0$
Continuous	Yes	Yes	Yes	Yes
Increasing		$\left(-\infty ,0\right)$		$\left(-\infty ,0\right)$
Decreasing	$\left(-\infty ,0\right)$, $\left(0,\infty \right)$	$\left(-\infty ,0\right)$	$\left(-\infty ,0\right)$, $\left(0,\infty \right)$	$\left(-\infty ,0\right)$
Symmetry	Origin	Y-axis	Origin	Y-axis
Bounded	Not bounded	Below	Not bounded	Below
Extrema	No	No	No	No
Asymptotes	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis	$x$ -axis, $y$ -axis
End behavior	$\underset{x\to \infty }{\lim }f\left(x\right)$	$\underset{x\to \infty }{\lim }g\left(x\right)$	$\underset{x\to \infty }{\lim }h\left(x\right)$	$\underset{x\to \infty }{\lim }k\left(x\right)$