Chapter 1.5, Problem 98E

To determine

To Describe: a way to identify a function as odd or even by inspecting the equation

Answer to Problem 98E

The graph of the function $f\left(x\right)=x^2-x^4$ using graphing utility is

  

Here the function is even function

The graph of $g\left(x\right)=2x^3+1$ using graphing utility is

  

Here the function neither even nor odd

The graph of function $h\left(x\right)=x^5-2x^3+x$ using graphing utility is

  

Here the function is odd

The graph of $j\left(x\right)=2-x^6-x^8$ using graphing utility

  

Here the function is even

The graph of $k\left(x\right)=x^5-2x^4+x-2$ using graphing utility

  

Here $k\left(x\right)=x^5-2x^4+x-2$ is neither even nor odd

The graph of function $p\left(x\right)=x^9+3x^5-x^3+x$ using graphing utility

  

Here the function is odd

A function, say $f\left(x\right)$ is called even function if $f\left(-x\right)=f\left(x\right)$ .

A function, say $f\left(x\right)$ is called odd function if $f\left(-x\right)=-f\left(x\right)$

The functions which doesn’t any of the above conditions is neither odd nor even

Explanation of Solution

Given:

The functions are:

  $\begin{array}{l}f\left(x\right)=x^2-x^{4}g\left(x\right)=2x^3+1\\ h\left(x\right)=x^5-2x^3+xj\left(x\right)=2-x^6-x^8\\ k\left(x\right)=x^5-2x^4+x-2p\left(x\right)=x^9+3x^5-x^3+x\end{array}$

Concept used:

An even function is symmetric about y-axis and the odd function is symmetric about origin.

The functions which are not symmetric about origin or y-axis are neither odd nor even.

Calculation:

The graph of the function $f\left(x\right)=x^2-x^4$ using graphing utility is

  

Here the function is symmetric about y-axis so the function is even.

Also

  $\begin{array}{c}f\left(-x\right)={\left(-x\right)}^2-{\left(-x\right)}^4\\ =x^2-x^4\\ =f\left(x\right)\end{array}$

Since $f\left(-x\right)=f\left(x\right)$ the function is even function

The graph of $g\left(x\right)=2x^3+1$ using graphing utility is

  

Here the functions is not symmetric about origin or y-axis so it is neither odd nor even.

Also

  $\begin{array}{c}g\left(-x\right)=2{\left(-x\right)}^3+1\\ =-2x^3+1\ne g\left(x\right)\text{or}-g\left(x\right)\end{array}$

Hence the function neither even nor odd

The graph of function $h\left(x\right)=x^5-2x^3+x$ using graphing utility is

  

Here the function is symmetric about origin so it is odd function.

Also

  $\begin{array}{c}h\left(-x\right)={\left(-x\right)}^5-2{\left(-x\right)}^3+\left(-x\right)\\ =-x^5+2x^3-x=-h\left(x\right)\end{array}$

Since $h\left(x\right)=-h\left(x\right)$ the function is odd

The graph of $j\left(x\right)=2-x^6-x^8$ using graphing utility

  

Here the function is symmetric about y-axis so it is even function

Also

  $\begin{array}{c}j\left(-x\right)=2-{\left(-x\right)}^6-{\left(-x\right)}^8\\ =2-x^6-x^8=j\left(x\right)\end{array}$

Since $j\left(-x\right)=j\left(x\right)$ the function is even

The graph of $k\left(x\right)=x^5-2x^4+x-2$ using graphing utility

  

Here the function is neither symmetric about origin nor y-axis hence it is neither odd nor even.

Also

  $\begin{array}{c}k\left(-x\right)={\left(-x\right)}^5-2{\left(-x\right)}^4+\left(-x\right)-2\\ =-x^5-2x^4-x-2\ne k\left(x\right)\text{or}-k\left(x\right)\end{array}$

Hence the $k\left(x\right)=x^5-2x^4+x-2$ is neither even nor odd

The graph of function $p\left(x\right)=x^9+3x^5-x^3+x$ using graphing utility

  

Here the function is symmetric about origin so it is an odd function

Also

  $\begin{array}{c}p\left(-x\right)={\left(-x\right)}^9+3{\left(-x\right)}^5-{\left(-x\right)}^3+\left(-x\right)\\ =-x^9-3x^5+x^3-x=-p\left(x\right)\end{array}$

Since $p\left(-x\right)=-p\left(x\right)$ the function is odd

Conclusion:

From all the above equations it can be concluded that

A function, say $f\left(x\right)$ is called even function if $f\left(-x\right)=f\left(x\right)$ .

A function, say $f\left(x\right)$ is called odd function if $f\left(-x\right)=-f\left(x\right)$

The functions which doesn’t any of the above conditions is neither odd nor even