Mathematics

This problem is really about understanding the range of a rational function by recognizing it as an equation you can solve for $x$. The key insight is to flip the perspective: instead of asking "what outputs does $f$ produce?", ask "for which values $y$ can I actually solve $f(x) = y$ for some real $x$?" Rearranging $y = \frac{x^2 - ax + 1}{x^2 + ax + 1}$ into a quadratic in $x$ and requiring real solutions will give you a constraint on $y$—that's your range. The parameter $a$ controls how "spread out" this range is.

This problem is really about understanding the range of a rational function by recognizing it as an equation you can solve for $x$. The key insight is to flip the perspective: instead of asking "what outputs does $f$ produce?", ask "for which values $y$ can I actually solve $f(x) = y$ for some real $x$?" Rearranging $y = \frac{x^2 - ax + 1}{x^2 + ax + 1}$ into a quadratic in $x$ and requiring real solutions will give you a constraint on $y$—that's your range. The parameter $a$ controls how "spread out" this range is.