Chapter 2.3, Problem 43AYU

In Problems 37-48, determine algebraically whether each function is even, odd, or neither.

47. $h\left(x\right)=\frac{-x^3}{3x^2-9}$

To determine

Check whether the function $f\left(x\right)=\frac{-x^3}{3x^2-9}$ is even, odd or neither.

Answer to Problem 43AYU

The given function $f$ is an odd function.

Explanation of Solution

By the definition of odd and even function,

“A function $f$ is even if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=f\left(x\right)$” and

“A function $f$ is odd if, for every number $x$ in its domain, the number $-x$ is also in the domain and $f\left(-x\right)=-f\left(x\right)$”.

Calculation:

Consider the function,

$f\left(x\right)=\frac{-x^3}{3x^2-9}$

Replace $x$ by $-x$,

$f\left(-x\right)=\frac{-{\left(-x\right)}^3}{3{\left(-x\right)}^2-9}=\frac{x^3}{3x^2-9}=-\frac{-x^3}{3x^2-9}=-f\left(x\right)$

From the statement, it can be concluded that the given function $f$ is an odd function.