Let f be a function for which f((x)/(3)) = x^2 + x + 1. Find the sum of all values of z for which f(3z) = 7.

2000 AMC 10 Problem 24 — Solution

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24.

Let $f$ be a function for which $f\left(\dfrac{x}{3}\right) = x^2 + x + 1.$ Find the sum of all values of $z$ for which $f(3z) = 7.$

$-\dfrac13$

$-\dfrac19$

$0$

$\dfrac59$

$\dfrac53$

To evaluate $f(3z),$ set $\dfrac{x}{3} = 3z,$ so $x = 9z.$ Then $\begin{aligned} f(3z) &= (9z)^2 + 9z + 1 \\ &= 81z^2 + 9z + 1. \end{aligned}$

Setting this equal to $7$ gives $81z^2 + 9z - 6 = 0.$

By the sum-of-roots formula, the sum of the values of $z$ is $-\dfrac{9}{81} = -\dfrac19.$

Thus, the correct answer is B.