Reciprocating the Roots of a Quintic

Note that $f\left(\dfrac{1}{x}\right)$ has roots that are the reciprocal of the roots of $f(x)$ . However, since this expression is not a polynomial, we must multiply it by $x^5$ to make it into a polynomial. Hence, $g(x)=x^5f\left(\dfrac{1}{x}\right)=-5x^5+3x^4-11x^3-2x^2+3x+5$ .

Thus, $|g(3)|=|-5(3)^5+3(3)^4-11(3)^3-2(3)^2+3(3)+5|=\boxed{1273}$ .

The problem did not specify that $f(x),g(x)$ are polynomial. However, I must assume that or else there are multiple answers.

Instead of "there is a bijection between the roots of $g(x)$ and the reciprocals of the roots of $f(x)$ ", the problem should say that the roots of $g(x)$ are the reciprocals of the roots of $f(x)$ . (If there exists a bijection between two finite sets, all that says is that the two sets contain the same number of elements.)