Chapter 5.4, Problem 52HP

To determine

To Find: The function is said to be odd if for every value of $x$ to obtain the domain, $-f\left(x\right)=f\left(-x\right)$ . Determine whether every odd degree polynomial function is also the odd function.

Answer to Problem 52HP

The statement is that every odd degree polynomial function is also the odd function is incorrect.

Explanation of Solution

The objective is to determine if an odd degree polynomial function is an odd function.

Consider that $n=2k+1$ and the function is,

  $f\left(x\right)=a_{2k+1}{x}^{2k+1}+a_{2}k{x}^{2k}+\mathrm{......}+a_{1}x+a_0$

For the function to be even $-f\left(x\right)=f\left(-x\right)$ then,

  $-a_{2k+1}{x}^{2k+1}+a_{2}k{x}^{2k}-\mathrm{.....}-a_{1}x+a_0=-a_{2k+1}{x}^{2k+1}-a_{2}k{x}^{2k}-\mathrm{.....}-a_{1}x-a_0$

Consider the odd degree of the polynomial function is only when,

  $\begin{array}{c}{a}_{2k}=a_{2k-2}\\ =a_2\\ =a_0\\ =0\end{array}$

The example for this is,

  $\begin{array}{l}f\left(x\right)=x^5-3x^3+x\text{odd}\\ g\left(x\right)=x^5+2x^4-3x^3+x\text{not}\text{odd}\end{array}$

Thus, the statement is that every odd degree polynomial function is also the odd function is incorrect.