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 12 Problem 15

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15.

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.

-\dfrac{1}{3}

-\dfrac{1}{9}

0

\dfrac{5}{9}

\dfrac{5}{3}

To evaluate

f(3z),

set

\dfrac{x}{3} = 3z,

x = 9z

This gives a quadratic in

z

; use the sum-of-roots formula

-\dfrac{b}{a}

Setting

\dfrac{x}{3} = 3z

gives

x = 9z,

\begin{aligned} f(3z) &= (9z)^2 + 9z + 1 \\ &= 81z^2 + 9z + 1 \\ &= 7. \end{aligned}

This rearranges to

81z^2 + 9z - 6 = 0.

By the sum-of-roots formula, the sum of the values of

z

-\dfrac{9}{81} = -\dfrac{1}{9}.

Thus, the correct answer is B.