Question

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

Short answer

The function is neither even nor odd.

Step-by-step solution

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.