Chapter 2.2, Problem 66E

Solution

Analyze the graph of $f\left(x\right)=x^{\pi },x^{\frac{1}{\pi }},x^{-\pi },-x^{\pi },-x^{\frac{1}{\pi }},-x^{-\pi }$ . Prepare a sketch of all six graphs on one set of axes, labelling each of the curves. Explain why each graph is positioned and shaped as it is.

  $f\left(x\right)=x^{\pi }$ , the graph looks like below graph because both k and a are greater than $1$ .

  $f\left(x\right)=x^{\frac{1}{\pi }}$ ,the graph looks like below graph because $k$ is greater than $1$ but $a$ lies between $0$ and $1$ .

  $f\left(x\right)=x^{-\pi }$ , the graph looks like below graph because $k$ is greater than $1$ but $a$ is lesser than $0$ .

  $f\left(x\right)=-x^{\pi }$ , the graph looks like below graph because $k$ is less than $0$ but $a$ is greater than $1$ .

  $f\left(x\right)=-x^{\frac{1}{\pi }}$ , the graph looks like below graph because $k$ is less than $0$ but $a$ lies between $0$ and $1$ .

  $f\left(x\right)=-x^{-\pi }$ , the graph looks like below graph because both $k$ and $a$ are less than $0$ .

Given:

  $f\left(x\right)=x^{\pi },x^{\frac{1}{\pi }},x^{-\pi },-x^{\pi },-x^{\frac{1}{\pi }},-x^{-\pi }$

Calculation:

The functions are $f\left(x\right)=x^{\pi },x^{\frac{1}{\pi }},x^{-\pi },-x^{\pi },-x^{\frac{1}{\pi }},-x^{-\pi }$

Below are the graphs for these functions.

  

Compare the given function to the standards function $f\left(x\right)=k{x}^a$ to tell why the shape of the graph is as shown above?

  $f\left(x\right)=x^{\pi }$ , the graph looks like above because both k and a are greater than $1$ .

  $f\left(x\right)=x^{\frac{1}{\pi }}$ ,the graph looks like above because $k$ is greater than $1$ but $a$ lies between $0$ and $1$ .

  $f\left(x\right)=x^{-\pi }$ , the graph looks like above because $k$ is greater than $1$ but $a$ is lesser than $0$ .

  $f\left(x\right)=-x^{\pi }$ , the graph looks like above because $k$ is less than $0$ but $a$ is greater than $1$ .

  $f\left(x\right)=-x^{\frac{1}{\pi }}$ , the graph looks like above because $k$ is less than $0$ but $a$ lies between $0$ and $1$ .

  $f\left(x\right)=-x^{-\pi }$ , the graph looks like above because both $k$ and $a$ are less than $0$ .