Q. 12 - Precalculus Enhanced with Graphing Utilities

Question

Is the function$f\left(x\right)=\frac{x^2}{2x+1}$ even, odd, or neither? 

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1. Given information.

Is the function $f\left(x\right)=\frac{x^2}{2x+1}$ even, odd, or neither? 

Step 2. Definition.

If the function is even then: 

$f\left(-x\right)=f\left(x\right)$

If the function is odd then: 

$f(-x)=-f\left(x\right)$

Step 3. Determine the function.

$$\begin{gathered} f\left(x\right)=\frac{x^2}{2x+1} \\ f\left(-x\right)=\frac{{\left(-x\right)}^2}{2\left(-x\right)+1}=\frac{x^2}{-2x+1}=-\frac{x^2}{2x-1} \end{gathered}$$

So, $f\left(-x\right)\ne f\left(x\right) \mathrm{and} f(-x)\ne -f\left(x\right)$.

The function is neither even nor odd.

Step 3. Conclusion.

The function is neither even nor odd.