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# The function f is defined for all positive integers n > 4 as

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Senior Manager
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The function f is defined for all positive integers n > 4 as [#permalink]

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15 Jan 2014, 12:00
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95% (hard)

Question Stats:

45% (01:43) correct 55% (01:45) wrong based on 386 sessions

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The function f is defined for all positive integers n > 4 as f(n) = 3n – 9 if n is odd and f(n) = 2n – 7 if n is even. What is the value of the positive integer a?

(1) f(f(a)) = a

(2) f(f(f(a))) is odd.
[Reveal] Spoiler: OA

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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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15 Jan 2014, 18:09
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Rock750 wrote:
The function f is defined for all positive integers n > 4 as f(n) = 3n – 9 if n is odd and f(n) = 2n – 7 if n is even. What is the value of the positive integer a?

(1) f(f(a)) = a

(2) f(f(f(a))) is odd.

I'm happy to help. As usual, this is a spectacularly clever problem from those folks at MGMAT.

Here's a blog you may find helpful on function notation:
http://magoosh.com/gmat/2012/function-n ... -the-gmat/

Notice that for this particular function, when f(n) has an even input, it yields an odd output, and vice versa: when it has an odd input, it yield an even output.

Statement #1:
Well, if a is even, then f(a) = 2a - 7, which will be odd, and f(f(a)) = 3(2a - 7) - 9 = 6a - 30.
Then, if f(f(a)) = a, we have
6a - 30 = a
5a = 30
a = 6
That's one possible value.
If a is odd, then f(a) = 3a - 9, which will be even, and f(f(a)) = 2(3a - 9) - 7 = 6a - 25
Then, if f(f(a)) = a, we have
6a - 25 = a
5a = 25
a = 5
That's also one possible value.
This statement yields two possible values, so no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Statement #2:
If f(f(f(a))) is odd,
then f(f(a)) is even,
and f(a) is odd,
and a is even.
This tells us that a is even, but a could be any even number.
This statement yields no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Combined
We have two values from statement #1. From statement #2, we know a must be even. This means that a = 6. Now, we can give a definitive answer to the prompt question. Combined, the statements are sufficient.

Does all this make sense?
Mike
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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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01 Feb 2014, 06:58
1
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Stmt1: f(f(a))=a

If a is odd f(a) = 3a-9 this is even.
f(f(a))=2(3a-9)-7 = a (given)
solving a = 5.

If a is even f(a) is odd.
solving f(f(a))=a
gives us a=6.

Since 2 values of a are possible, stmt 1 is INSUFF.

Stmt2: f(f(f(a))) is odd
then f(f(a)) is even
f(a) is odd
a is even. Clearly INSUFF

Stmt1+stmt2: a=5 or 6 and a is even. a can only be 6. Hence C.
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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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11 May 2014, 14:59
mikemcgarry wrote:
Rock750 wrote:
The function f is defined for all positive integers n > 4 as f(n) = 3n – 9 if n is odd and f(n) = 2n – 7 if n is even. What is the value of the positive integer a?

(1) f(f(a)) = a

(2) f(f(f(a))) is odd.

I'm happy to help. As usual, this is a spectacularly clever problem from those folks at MGMAT.

Here's a blog you may find helpful on function notation:
http://magoosh.com/gmat/2012/function-n ... -the-gmat/

Notice that for this particular function, when f(n) has an even input, it yields an odd output, and vice versa: when it has an odd input, it yield an even output.

Statement #1:
Well, if a is even, then f(a) = 2a - 7, which will be odd, and f(f(a)) = 3(2a - 7) - 9 = 6a - 30.
Then, if f(f(a)) = a, we have
6a - 30 = a
5a = 30
a = 6
That's one possible value.
If a is odd, then f(a) = 3a - 9, which will be even, and f(f(a)) = 2(3a - 9) - 7 = 6a - 25
Then, if f(f(a)) = a, we have
6a - 25 = a
5a = 25
a = 5
That's also one possible value.
This statement yields two possible values, so no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Statement #2:
If f(f(f(a))) is odd,
then f(f(a)) is even,
and f(a) is odd,
and a is even.
This tells us that a is even, but a could be any even number.
This statement yields no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Combined
We have two values from statement #1. From statement #2, we know a must be even. This means that a = 6. Now, we can give a definitive answer to the prompt question. Combined, the statements are sufficient.

Does all this make sense?
Mike

Hi Mike,

is there a reason why choosing numbers here doesn't work or isn't optimal?

I tried use 5,6 and 7,8 and my values are all over?

Thanks

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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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11 May 2014, 22:27
russ9 wrote:
Hi Mike,

is there a reason why choosing numbers here doesn't work or isn't optimal?

I tried use 5, 6 and 7, 8 and my values are all over?

Thanks

Dear russ9
Think about the prompt question, "What is the value of a?" It may be that a has just one value, or more than one. If you find one value, that's absolutely no guarantee that there aren't other values that also work. Suppose, for the sake of argument, that the two values that worked were a = 6 and a = 50 --- plugging in numbers for some single digit cases would never tell you that there's more than one answer. Do you see what I mean?

Remember, GMAT DS is NOT about "find the answer" --- it's more about "is it possible to find a unique and sensible answer?" If you were looking for one and only one answer, then plugging in numbers would make sense --- that might not be so bad on GMAT PS. But on GMAT DS, that misses the point in a problem such as this.

Does this make sense?
Mike
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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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15 May 2014, 17:22
mikemcgarry wrote:
russ9 wrote:
Hi Mike,

is there a reason why choosing numbers here doesn't work or isn't optimal?

I tried use 5, 6 and 7, 8 and my values are all over?

Thanks

Dear russ9
Think about the prompt question, "What is the value of a?" It may be that a has just one value, or more than one. If you find one value, that's absolutely no guarantee that there aren't other values that also work. Suppose, for the sake of argument, that the two values that worked were a = 6 and a = 50 --- plugging in numbers for some single digit cases would never tell you that there's more than one answer. Do you see what I mean?

Remember, GMAT DS is NOT about "find the answer" --- it's more about "is it possible to find a unique and sensible answer?" If you were looking for one and only one answer, then plugging in numbers would make sense --- that might not be so bad on GMAT PS. But on GMAT DS, that misses the point in a problem such as this.

Does this make sense?
Mike

Makes total sense. Thanks, Mike!

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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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11 Jun 2015, 04:55
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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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19 Jun 2016, 02:00
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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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03 Jul 2017, 06:17
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Re: The function f is defined for all positive integers n > 4 as [#permalink]

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03 Jul 2017, 08:31
Given: f(n) = 3n-9 when n is odd
--> f(n) = O*O - O = O - O =E
--> f(n) is even when n is odd

F(n) = 2n -7 when n is even
--> f(n) = e* e - o = e - o = o
--> f(n) is odd when n is even

f(n) changes n from even to odd and from odd to even

1. f(f(a) = a

when a is even
f(f(a)) = 3(2a - 7) - 9
= 6a -21 -9 = 6a - 30

when a is odd
f(f(a)) = 2 (3a - 9) - 7
=6a - 18 - 7 =6a - 25

Depending on the odd/even nature of a, value of function changes. Thus, insuff.

2. f(f(f(a))) is odd. This tells us whether a is even or odd. Insuff.

1 & 2 :
we now now which formula to use from stmt 1 --> suff.

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Re: The function f is defined for all positive integers n > 4 as   [#permalink] 03 Jul 2017, 08:31
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