Equation of function

Question

How can I find equation for this function?

the asymptotes on the sides can be for example at $x = -8$ and $x = 8.$

Answer 1

Let's give it a try: since the function we look for $\,f(x)\,$ is even (according to the given graph), we can begin with $\,g(x)=(x^2-4)(x^2-9)(x^2-25)(x^2-36)(x^2-49)\,$ , and it has vertical asymptotes at $\,x=0\,,\,\pm 8\,$ , we can put

$$k(x)=\frac{g(x)}{x^2(64-x^2)}$$

(not $\,x^2-64\,$ in the denominator as we need, for example $\,f(x)\xrightarrow [x\to -8^-]{} -\infty\,$

and since we need

$$\lim_{x\to \pm\infty}f(x)=0$$

we need the denominator's degree higher than the numerator's but in a way as to be sensible to sign changes around $\,\pm\,8\,$, so for example

$$f(x):=\frac{g(x)}{x^2(64-x^2)^{9}}$$

Check carefully whether the above makes the cut.