Let f be a function for which f(x/3) = x^2 +… | GMAT Sprint

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PS · Problem Solving

705-805

Functions

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

Correct answer marked below

-\frac{1}{3}

-\frac{1}{9}

Correct

0

\frac{5}{9}

\frac{5}{3}

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Explanation

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Official Explanation

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We are given that

f(x/3) = x^2 + x + 1

.

We want to find z such that

f(3z) = 7

.

Notice that

3z = \frac{9z}{3}

, so we can set

x = 9z

in the given functional equation:

f(3z) = f\left(\frac{9z}{3}\right) = (9z)^2 + 9z + 1

.

Set this equal to 7:

(9z)^2 + 9z + 1 = 7

81z^2 + 9z + 1 = 7

81z^2 + 9z - 6 = 0

Divide by 3 to simplify:

27z^2 + 3z - 2 = 0

We can factor or use the quadratic formula. Factoring:

(9z + 3)(9z - 2) = 0

Thus either:

9z + 3 = 0 \quad \Rightarrow \quad z = -\frac{3}{9} = -\frac{1}{3}

…

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What this question tests

Your ability to evaluate and compose functions, including unusual defined operations.

Applying function operations in the wrong order or ignoring domain restrictions.