Finding a function with given conditions I want to find the equation of a function $f(x)$ such that the following conditions are met:
$(i)\ f(3) = 0, f(0) =1$.
$(ii)\ f(x)$ is an even function.
$(iii)\ f(x)$ has vertical asymptodes at $x=\pm 4$.
$(iv)\ f(x)$ has horizontal asymptode at $ y=2$.
Now since the function is even, I get $x=-3$ as another root and then I can graph the function. But how do I find its equation? Any hints? Thank you.
In general is there some method or software to find the equation of a function given its graph?

We use building blocks having one or more of the wanted properties and put them together conveniently so that the final function $f$ has all wanted properties.

  
*
  
*We start with (iii) and take as function with asymptote at $x=\pm 4$ the rational function 
  \begin{align*}
x\to\frac{1}{(x-4)(x+4)}=\frac{1}{x^2-16}\tag{1}
\end{align*}
which is already even.

To get a zero at $x=3$ we multiply with $x-3$ and to keep the function even we multiply with $(x-3)(x+3)=x^2-9$.

  
*
  
*We obtain
  \begin{align*}
x\to\frac{x^2-9}{x^2-16}\tag{2}
\end{align*}

Note that $\frac{x^2-9}{x^2-16}$ evaluated at $x=0$ is $\frac{9}{16}$. 

  
*
  
*We multiply the function (2) with $\frac{16}{9}$ and obtain with
  \begin{align*}
x\to\frac{16}{9}\cdot\frac{x^2-9}{x^2-16}\tag{3}
\end{align*}
  an even function which fulfills (i) and (ii).
  

Finally we extend (3) to get a function with horizontal asymptote at $y=2$. Note that  multiplying the function with $e^{-x^2}$ gives 
\begin{align*}
\lim_{x\to\pm\infty}\frac{x^2-9}{x^2-16}e^{-x^2}=0
\end{align*}
Since we also have
\begin{align*}
\lim_{x\to\pm\infty}\frac{x^2-9}{x^2-16}=1
\end{align*}
we          use  the  Ansatz
\begin{align*}
f(x)=\frac{16}{9}\cdot\frac{x^2-9}{x^2-16}\left(ae^{-x^2}+b\right)
\end{align*}
solve
\begin{align*}
f(0)&=a+b=1\\
\lim_{x\to\pm\infty}f(x)&=\frac{16}{9}b=2
\end{align*}
and  find $a=-\frac{1}{8}$ and $b=\frac{9}{8}$.

  
*
  
*We finally obtain a function
\begin{align*}
&\color{blue}{f:\mathbb{R}\to\mathbb{R}\setminus\{\pm 4\}}\\
&\color{blue}{f(x)=\frac{2}{9}\cdot\frac{x^2-9}{x^2-16}\left(-e^{-x^2}+9\right)}
\end{align*}
  which fulfills (i) to (iv).
  

Note: To find an even function is usually not a problem. When we have a function $g$ fulfilling all other properties we can take $f(x)=\frac{1}{2}\left(g(x)+g(-x)\right)$ to get an even function with the wanted properties.
