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

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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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Difficulty:

95% (hard)

Question Stats:

41% (02:51) correct 59% (02:55) wrong based on 343 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.

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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
28
11
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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##### General Discussion
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Joined: 25 Oct 2013
Posts: 143
Re: The function f is defined for all positive integers n > 4 as  [#permalink]

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01 Feb 2014, 06:58
1
2
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
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4487
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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Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
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Posts: 239
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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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]

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08 Jan 2018, 07:04
Refer to the pic attached.

Attachments

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

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08 Jan 2018, 07:19
it took me 3 min.
After I have found the answer, I note that st 1 helps to solve the st 2.
In other words, st 1 gives value while st 2 only limits to one certain value.

VeritasPrepKarishma, chetan2u IMPORTANT: IF THERE IS NO VALUE FOR A, THEN E IS THE ANSWER. (am i correct? )
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Re: The function f is defined for all positive integers n > 4 as  [#permalink]

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08 Jan 2018, 09:12
1
chesstitans wrote:
it took me 3 min.
After I have found the answer, I note that st 1 helps to solve the st 2.
In other words, st 1 gives value while st 2 only limits to one certain value.

VeritasPrepKarishma, chetan2u IMPORTANT: IF THERE IS NO VALUE FOR A, THEN E IS THE ANSWER. (am i correct? )

Hi..

yes, if there are no value from statement I , ans will be E..
and that could happen if say the value has $$a^2$$ instead of a that is $$F(a)=3a^2-4$$... then you may land up with more than 2 values from statement 1...

Just two points which will help in answering the Q
1) the function is LINEAR, so it will give you one value
2) F(odd) is even and F(even) is odd...

if you are able to get to above two points, you will understand that there can be ONLY one value if x is ODD and one value when x is even.
and statement II will tell you whether a is odd or even..
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Re: The function f is defined for all positive integers n > 4 as  [#permalink]

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03 Mar 2018, 06:18
Hi.

For many similar problems, I can solve them but m not able to do it under two minutes. Any advice?

Thanks

Posted from my mobile device
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Posts: 56257
Re: The function f is defined for all positive integers n > 4 as  [#permalink]

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05 Mar 2018, 01:20
Mudit27021988 wrote:
Hi.

For many similar problems, I can solve them but m not able to do it under two minutes. Any advice?

Thanks

Posted from my mobile device

13. Functions

5. Strategies and Tactics To Increase Your Score

6. Strategies and Tactics To Speedd-Up

[/list]

For more check below:
ALL YOU NEED FOR QUANT ! ! !

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

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11 Mar 2019, 06:48
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,

What is the reason behind the The function f is defined for all positive integers n > 4... part?

I'm not able to understand it.

Thanks!
Re: The function f is defined for all positive integers n > 4 as   [#permalink] 11 Mar 2019, 06:48
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