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Q 43.

Expert-verified
Found in: Page 90

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 33– 44, determine algebraically whether each function is even, odd, or neither.$h\left(x\right)=\frac{-{x}^{3}}{3{x}^{2}-9}$

The given function $h\left(x\right)=\frac{-{x}^{3}}{3{x}^{2}-9}$ is an odd function.

See the step by step solution

## Step 1. Write the given information and use the definition of odd and even function.

The given function is:

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

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

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

## Step 2. Determine if the function is even.

Replace $x$ by $-x$ in the given function,

$h\left(-x\right)=\frac{-{\left(-x\right)}^{3}}{3{\left(-x\right)}^{2}-9}\phantom{\rule{0ex}{0ex}}=\frac{-{\left(-x\right)}^{3}}{3{\left(x\right)}^{2}-9}\phantom{\rule{0ex}{0ex}}=\frac{{x}^{3}}{3{x}^{2}-9}$

Since $h\left(-x\right)\ne h\left(x\right),h$ is not an even function.

## Step 3. Determine if the function is odd.

Find the function $-h\left(x\right)$,

role="math" localid="1645811470834" $-h\left(x\right)=-\left(\frac{-{\mathrm{x}}^{3}}{3{\mathrm{x}}^{2}-9}\right)\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{x}}^{3}}{3{\mathrm{x}}^{2}-9}$

Since $h\left(-x\right)=-h\left(x\right),h$ is an odd function.