Chapter 1, Problem 20RE

To determine

To check: Whether the function $y=x^5-x^3-x$ is even, odd or neither.

Answer to Problem 20RE

The function $y=x^5-x^3-x$ is an odd function.

Explanation of Solution

Given information: The given function is $y=x^5-x^3-x$ .

Calculation:

Even function: A function $f\left(x\right)$ is said to be an even function if it satisfies the condition $f\left(-x\right)=f\left(x\right)$ .

Odd function: A function $f\left(x\right)$ is said to be an odd function if it satisfies the condition $f\left(-x\right)=-f\left(x\right)$ .

Substitute $-x$ for $x$ in the given function $y=f\left(x\right)=x^5-x^3-x$ .

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

Therefore, the function $y=x^5-x^3-x$ is an odd function.