Q. 7 - Precalculus Enhanced with Graphing Utilities

Question

Determine whether the function $g\left(x\right)=\frac{2x^3}{x^4+1}$ is even, odd, or neither. Is the graph of $g$ symmetric with respect to the $x$-axis, $y$-axis, or origin?

Step-by-Step Solution

Answer

The function $g\left(x\right)=\frac{2x^3}{x^4+1}$ is odd. Since the given function is odd, it is symmetric with respect to the origin.

Step 1 Given function is,

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

Substitute $x=-x$ in the given equation.

The function is even if $g\left(-x\right)=g\left(x\right)$ and odd it $g\left(-x\right)=-g\left(x\right)$. If these conditions are not satisfied, then the function is neither odd nor even.

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

So, the given function is odd.

Step 2 Determine whether the graph of the given function is symmetric with respect to x -axis or y -axis or the origin.

When the function is even, then the graph of that function will be symmetric with respect to the $y$-axis. When the function is odd, then the graph of that function will be symmetric with respect to the origin.

Since the given function is odd,  the graph of that function will be symmetric with respect to the origin.