Chapter 9.3, Problem 37PPS

a.

To determine

To make a table of values comparing the two functions.

Answer to Problem 37PPS

The table comparing the values of the given functions is:

  

Explanation of Solution

Given information:

The given functions are:

  $f\left(x\right)=\frac{1}{x}\text{ and }g\left(x\right)=\frac{1}{x^2}$

Calculation:

The table comparing the values of the given functions is:

  

b.

To determine

To use the table of values to graph both the functions.

Explanation of Solution

Given information:

The given functions are:

  $f\left(x\right)=\frac{1}{x}\text{ and }g\left(x\right)=\frac{1}{x^2}$

Graph:

The graph of the functions are:

  

c.

To determine

To compare and contrast the two graphs.

Answer to Problem 37PPS

The positive portion in the graph of $g\left(x\right)=\frac{1}{x^2}$ is similar to the graph of $f\left(x\right)=\frac{1}{x}$ . Positive values of $x$ produce positive values of $f\left(x\right)$ . The negative portion of $g\left(x\right)=\frac{1}{x^2}$ appears to be the reflection of $f\left(x\right)=\frac{1}{x}$ in the x-axis. Negative values of $x$ produce positive values of $g\left(x\right)$ .

Explanation of Solution

Given information:

The given functions are:

  $f\left(x\right)=\frac{1}{x}\text{ and }g\left(x\right)=\frac{1}{x^2}$

The positive portion in the graph of $g\left(x\right)=\frac{1}{x^2}$ is similar to the graph of $f\left(x\right)=\frac{1}{x}$ . Positive values of $x$ produce positive values of $f\left(x\right)$ . The negative portion of $g\left(x\right)=\frac{1}{x^2}$ appears to be the reflection of $f\left(x\right)=\frac{1}{x}$ in the x-axis. Negative values of $x$ produce positive values of $g\left(x\right)$ .

d.

To determine

To make a conjecture about the difference between the graphs of functions of the form $f\left(x\right)=\frac{1}{x^n}$ with an even exponent in the denominator and those with an odd exponent in the denominator.

Answer to Problem 37PPS

When $n$ is even, the graph will show symmetry with respect to the y-axis.

When $n$ is odd, the graph will show symmetry with respect to the origin.

Explanation of Solution

Given information:

The given functions are:

  $f\left(x\right)=\frac{1}{x}\text{ and }g\left(x\right)=\frac{1}{x^2}$

When $n$ is even, the graph will show symmetry with respect to the y-axis.

When $n$ is odd, the graph will show symmetry with respect to the origin.