Answer
Graph the function as:
Work Step by Step
Simplify the above equation:
$\begin{align}
& \frac{x}{\left( 2x+6 \right)}-\frac{9}{{{x}^{2}}-9}=\frac{x}{2\left( x+3 \right)}-\frac{9}{{{x}^{2}}-{{\left( 3 \right)}^{2}}} \\
& =\frac{x}{2\left( x+3 \right)}-\frac{9}{\left( x-3 \right)\left( x+3 \right)} \\
& =\frac{x\left( x-3 \right)-9\times 2}{2\left( x-3 \right)\left( x+3 \right)}
\end{align}$
Solve:
$\begin{align}
& \frac{x\left( x-3 \right)-9\times 2}{2\left( x-3 \right)\left( x+3 \right)}=\frac{{{x}^{2}}-3x-18}{2\left( x-3 \right)\left( x+3 \right)} \\
& =\frac{{{x}^{2}}-6x+3x-18}{2\left( x-3 \right)\left( x+3 \right)} \\
& =\frac{x\left( x-6 \right)+3\left( x-6 \right)}{2\left( x-3 \right)\left( x+3 \right)} \\
& =\frac{\left( x+3 \right)\left( x-6 \right)}{2\left( x-3 \right)\left( x+3 \right)}
\end{align}$
Simplify further:
$\frac{\left( x+3 \right)\left( x-6 \right)}{2\left( x-3 \right)\left( x+3 \right)}=\frac{\left( x-6 \right)}{2\left( x-3 \right)}$
Hence, the simplified expression is $\frac{\left( x-6 \right)}{2\left( x-3 \right)}$.
Hence, the equation for function $f$ is $f\left( x \right)=\frac{\left( x-6 \right)}{2\left( x-3 \right)}.$
Substitute $x$ equal to $-x$ and find the symmetry of the function.
$\begin{align}
& f\left( -x \right)==\frac{\left( -x-6 \right)}{2\left( -x-3 \right)} \\
& =\frac{-\left( x+6 \right)}{-2\left( x+3 \right)} \\
& =\frac{\left( x+6 \right)}{2\left( x+3 \right)}
\end{align}$
The obtained value of $f\left( -x \right)$ is not equal to either function $f\left( x \right)$ or $-f\left( x \right)$. Therefore, the graph of the function is not symmetric with respect to the y- axis nor the origin.
To find the y-intercept, evaluate $f\left( 0 \right)$ such that
$\begin{align}
& f\left( 0 \right)=\frac{\left( 0-6 \right)}{2\left( 0-3 \right)} \\
& =\frac{-6}{-6} \\
& =1
\end{align}$
Hence, the y-intercept is $1$ , so the graph passes through $\left( 0,1 \right)$.
To find the x-intercept, put the numerator equal to 0, that is, $p\left( x \right)=0$.
$\left( x-6 \right)=0$
Add $6$ to both sides of the equation:
$\begin{align}
& x+6-6=6 \\
& x=6
\end{align}$
Thus, the x-intercept is $6$ and the graph passes through the point $\left( 6,0 \right)$. Now, find the vertical asymptotes:
To find the vertical asymptotes, put the denominator of the function equal to zero.
$2\left( x-3 \right)=0$
Divide both sides of the equation with 2:
$\begin{align}
& \frac{2\left( x-3 \right)}{2}=\frac{0}{2} \\
& x-3=0
\end{align}$
Add 3 to both the sides of the equation:
$\begin{align}
& x-3+3=0+3 \\
& x=3
\end{align}$
Thus, the graph has a vertical asymptote and its equation is $x=3$.
Find horizontal asymptotes:
To find the horizontal asymptotes, if the degree of the denominator and numerator is the same, then divide the leading coefficient of the numerator by the leading coefficient of the denominator.
The leading coefficient of the numerator and denominator are 1 and 2, respectively.
Therefore, the equation's horizontal asymptote is $y=\frac{1}{2}$.
