4. (9th grade) Among the values of the function

$$
f(x)=\frac{\left(x^{2}+1\right)^{2}}{x\left(x^{2}-1\right)}
$$

find those that it takes the most number of times?
Answer: $(-\infty,-4) \cup(4,+\infty)$.

Solution. The desired values are precisely those $a$ for which the equation has the maximum number of roots:

$$
\begin{gathered}
f(x)=a \Longleftrightarrow\left(x^{2}+1\right)^{2}-a x\left(x^{2}-1\right)=0(\Rightarrow x \neq 0, \pm 1) \Longleftrightarrow\left(x^{2}-1\right)^{2}-a x\left(x^{2}-1\right)+4 x^{2}=0 \Longleftrightarrow \\
\Longleftrightarrow \phi_{a}(t):=t^{2}-a t+4=0, \text { where } t=\frac{x^{2}-1}{x}
\end{gathered}
$$

Since the equation $t=\frac{x^{2}-1}{x} \Longleftrightarrow x^{2}-t x-1=0$ has exactly two roots for any $t$, the original equation has the maximum (equal to 4) number of roots precisely when the quadratic polynomial $\phi_{a}$ has two roots, i.e., when $a^{2}-16>0 \Longleftrightarrow|a|>4$.