Answer
Graph the function as:
Work Step by Step
Step 1: Substitute $x=-x$
$\begin{align}
& f\left( x \right)=\frac{4{{x}^{2}}}{{{x}^{2}}+1} \\
& f\left( -x \right)=\frac{4{{\left( -x \right)}^{2}}}{{{\left( -x \right)}^{2}}+1} \\
& =\frac{4{{x}^{2}}}{{{x}^{2}}+1} \\
& =f\left( x \right)
\end{align}$
The function $f\left( -x \right)$ is equal to $f\left( x \right)$. Hence, the graph of the function is symmetric about the $y$ -axis.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$.
$\begin{align}
& \frac{4{{x}^{2}}}{{{x}^{2}}+1}=0 \\
& x=0
\end{align}$ ,
Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$.
$f\left( 0 \right)=0$
Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is:
$y=4$
Step 5: For the vertical asymptote, equate the denominator to 0.
The function has no real asymptotes.
