If \(f(g(3)) = 12\), what is \(f(g(4))\)?


(1) \(g(x) = 2x + 2\)

(2) \(f(g(5)) = 18\)
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The data is sufficient if we can find a unique value for f(g(4)).
From question stem we know f(g(3)) = 12.

Statement 1:\(g(x) = 2x + 2\)
We will be able to compute g(3)to be 8 and g(4) to be 10

f(g(3)) = f(8) = 12.
f(g(4)) = f(10) = ?

It coule be any value depending on what f(x) is. Here are two such values it can take.
If f(x) = 1.5x, f(8) = 12 and f(10) = 15
Alternatively, if f(x) = x + 4, f(8) = 12 and f(10) = 14.

Without additional information about f(x), we cannot find a unique value for f(g(4))

Statement 1 alone is NOT sufficient.

Statement 2: \(f(g(5)) = 18\)
Without knowing what g(x) is and what f(x), we will not be able extrapolate with this information and the question stem.
For instance, if f(g(x)) = 3x + 3, then f(g(3)) = 12 and f(g(5)) = 18. And f(g(4)) = 15
Alternatively, if f(g(x)) = sqrt(1.5) * f(g(x - 1)), f(g(5)) = sqrt(1.5) * f(g(4)) = sqrt(1.5) * sqrt(1.5) f(g(3)) = 1.5 * 12 = 18 and f(g(4)) = sqrt(1.5) * 12 which is not 15.

Statement 2 alone is not sufficient.

Statements together: g(x) = 2x + 2 and f(g(5)) = 18
g(3) = 8, g(4) = 10 and g(5) = 12
For instance, f(x) = 1.5x. If so, f(g(3)) = f(8) =12; f(g(5)) = f(12) = 18 and f(g(4)) = f(10) = 15
Alternatively, f(x) = sqrt (1.5)f(x - 2). If so, f(g(5)) = f(12) = sqrt(1.5)*f(10) = sqrt(1.5) * sqrt(1.5) * f(8) = 1.5 * 12 = 18
And f(g(4)) = f(10) = sqrt(1.5) * f(8) = sqrt(1.5) * 12 which is not 15.

Statements together not sufficient. Choice E is the answer.
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