Good day! I encounter a problem on rational function wherein I almost satisfy all the given requirements to derive the rational function except the last one. Any idea would be of great help, Thanks!
Find the function satisfying the following:
- $f(3) = 0$
- $f(x) = f(-x)$
- Vertical Asymptotes $x = 4$ and $x = -4$
- Horizontal Asymptotes $y = 2$
- $f(0) = 1$
Hint. It is easy to verify that the following rational map has properties 2. 3. 4. $$h(x)=\frac{2x^2}{(x^2-16)}.$$ Now let us find a rational function $g$ such that $f=h+g$ has all the requested properties. For $g$ we need an even rational functions which has no vertical asymptotes and goes to $0$ as $x\to \pm \infty$. For example we can try something like $$g(x)=\frac{A}{1+x^2}+\frac{B}{2+x^2}$$ where the constants $A$ and $B$ have to be determined such that $f(3)=0$ and $f(0)=1$. So it remains to solve the linear system $$\begin{cases} \frac{A}{10}+\frac{B}{11}=0-h(3)\\ A+\frac{B}{2}=1-h(0) \end{cases}$$ which has a unique solution (the same procedure works for the more general situation where $f(3)=a$ and $f(0)=b$).
P.S. We expect "several" solutions for this problem.