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Bunuel
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For all integers n, the function f is defined by f(n)=(a)^6/n where 11 May 2022, 05:00
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

35% (medium)

Question Stats:

63% (01:15) correct 37% (01:17) wrong based on 52 sessions

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For all integers \(n,\) the function \(f\) is defined by \(f(n)=(a)^{\frac{6}{n}}\) where \(a\) is a constant. What is the value of \(f(1)\)?

(1) \(f(2)=64\)

(2) \(f(3)=16\)
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BrentGMATPrepNow
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For all integers n, the function f is defined by f(n)=(a)^6/n where 14 May 2022, 06:53
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Bunuel
For all integers \(n,\) the function \(f\) is defined by \(f(n)=(a)^{\frac{6}{n}}\) where \(a\) is a constant. What is the value of \(f(1)\)?

(1) \(f(2)=64\)

(2) \(f(3)=16\)
Given: \(f(n)=(a)^{\frac{6}{n}}\) where \(a\) is a constant.

Target question: What is the value of \(f(1)\)?

Statement 1: \(f(2)=64\)
Plug \(n = 2\) into the given function to get: \(f(2)=(a)^{\frac{6}{2}} = 64\)
Simplify: \((a)^{3} = 64\)
So, it must be the case that \(a = 4\), which means our function is defined as follows: \(f(n)=(4)^{\frac{6}{n}}\)
So, \(f(1)=(4)^{\frac{6}{1}} = 4^6 = 4096\) [On test day, I would never waste time actually calculating 4^6, since all we need to do is determine whether there's only one answer to the target question]
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: \(f(3)=16\)
Plug \(n = 3\) into the given function to get: \(f(2)=(a)^{\frac{6}{3}} = 16\)
Simplify: \((a)^{2} = 16\)
There are two possible solutions to this equation: \(a = 4\) and \(a = -4\)

Important: Some students will assume that since we have two possible values for \(a\), statement 2 must be insufficient. However, the target question doesn't ask us for the value of \(a\); it asks us for the value of \(f(1)\). So let's see what happens for each possible a-value.

Case a: \(a = 4\). In this case, \(f(1)=(4)^{\frac{6}{1}} = 4^6 = 4096\)

Case b: \(a = -4\). In this case, \(f(1)=(-4)^{\frac{6}{1}} = (-4)^6 = 4096\)

Aha!! We get the same answer to the target question in both cases, which means it must be the case that \(f(1)=4096\)
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent
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Bunuel
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