Chapter 1, Problem 23RE

To determine

To check: Whether the function $y=\frac{x^4+1}{x^3-2x}$ is even, odd or neither.

Answer to Problem 23RE

The function $y=\frac{x^4+1}{x^3-2x}$ is an odd function.

Explanation of Solution

Given information: The given function is $y=\frac{x^4+1}{x^3-2x}$ .

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 $y=f\left(x\right)=\frac{x^4+1}{x^3-2x}$ .

  $\begin{array}{c}f\left(-x\right)=\frac{{\left(-x\right)}^4+1}{{\left(-x\right)}^3-2\left(-x\right)}\\ =\frac{x^4+1}{-x^3+2x}\\ =-\left(\frac{x^4+1}{x^3-2x}\right)\\ =-f\left(x\right)\end{array}$

Therefore, the function $y=\frac{x^4+1}{x^3-2x}$ is an odd function.