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

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:
https://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.
Answer = (C)

Does all this make sense?
Mike
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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.

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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Makes total sense. Thanks, Mike!

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.
Answer is C

Refer to the pic attached.

answer is C

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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Hi.

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

m Looking to score Q50 or above. So please advice accordingly.


Thanks

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Hope it helps.
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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!

This is an old question, but this thread popped up and I thought I'd answer it for anybody who's reading this now.

"defined for all positive integers n>4" means that you're only allowed to plug certain values of n into the function. For other values, the function is "undefined," which means it has no value. For example, you couldn't test f(0) as part of solving this particular problem, because, since 0 is less than 4, f(0) actually doesn't have a value at all.

Many, but not all, functions are only defined for certain inputs. For instance, you may have a function that's only defined for positive values, or for integers, or for numbers less than 100.
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