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# The function g(x) is defined for integers x such that if x

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The function g(x) is defined for integers x such that if x [#permalink]  15 Jan 2013, 15:46
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20% (03:51) correct 79% (02:46) wrong based on 209 sessions
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?

A. 1
B. 5
C. 7
D. 8
E. 11
[Reveal] Spoiler: OA
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Re: The function g(x) is defined for integers x such that if x [#permalink]  16 Jan 2013, 00:50
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MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?
a)1 b)5 ,c)7 ,d)8 ,e)11

Wow, more like a mathmatical puzzle than a gmat question. I love it!

Let me define terms:
in g(x) = R
x is argument, R is result, g() is function,
in g(g(g(g(g(x))))), g1 is inner most, g5 is outermost for identification.

From definition of function g, we can deduce that:
If Result is even then two possibilities for argument = 1 Even 1 Odd
If Result is odd then one possibility for argument = 1 Even

Since final result = 19 = Odd

Possibilities:
g1: 1 Even
g2: 1*(Even,Odd ) = 1 Even 1 Odd
g3: 1*(Even,Odd) + 1 Even = 2 Even 1 Odd
g4: 2*(Even, Odd) + 1 Even = 3 Even 2 Odd
g5: 3*(Even, Odd) + 2 Even = 5 Even 3 Odd = Total 8

Ans D it is!
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Re: The function g(x) is defined for integers x such that if x [#permalink]  16 Jan 2013, 00:58
7
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Expert's post
MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?
a)1 b)5 ,c)7 ,d)8 ,e)11

Hope this image helps you clarify these possible 8 set of values of x.

-Shalabh Jain
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Re: The function g(x) is defined for integers x such that if x [#permalink]  16 Jan 2013, 01:18
Vips0000 wrote:
MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?
a)1 b)5 ,c)7 ,d)8 ,e)11

Wow, more like a mathmatical puzzle than a gmat question. I love it!

Let me define terms:
in g(x) = R
x is argument, R is result, g() is function,
in g(g(g(g(g(x))))), g1 is inner most, g5 is outermost for identification.

From definition of function g, we can deduce that:
If Result is even then two possibilities for argument = 1 Even 1 Odd
If Result is odd then one possibility for argument = 1 Even

Since final result = 19 = Odd

Possibilities:
g1: 1 Even
g2: 1*(Even,Odd ) = 1 Even 1 Odd
g3: 1*(Even,Odd) + 1 Even = 2 Even 1 Odd
g4: 2*(Even, Odd) + 1 Even = 3 Even 2 Odd
g5: 3*(Even, Odd) + 2 Even = 5 Even 3 Odd = Total 8

Ans D it is!

Vips im totally lost in this... can u explain!!!
how u started g1 with even? based on answer choices?
if so how come u calculated g2?
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Re: The function g(x) is defined for integers x such that if x [#permalink]  16 Jan 2013, 01:42
5
KUDOS
shanmugamgsn wrote:
Vips0000 wrote:
MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?
a)1 b)5 ,c)7 ,d)8 ,e)11

Wow, more like a mathmatical puzzle than a gmat question. I love it!

Let me define terms:
in g(x) = R
x is argument, R is result, g() is function,
in g(g(g(g(g(x))))), g1 is inner most, g5 is outermost for identification.

From definition of function g, we can deduce that:
If Result is even then two possibilities for argument = 1 Even 1 Odd
If Result is odd then one possibility for argument = 1 Even

Since final result = 19 = Odd

Possibilities:
g1: 1 Even
g2: 1*(Even,Odd ) = 1 Even 1 Odd
g3: 1*(Even,Odd) + 1 Even = 2 Even 1 Odd
g4: 2*(Even, Odd) + 1 Even = 3 Even 2 Odd
g5: 3*(Even, Odd) + 2 Even = 5 Even 3 Odd = Total 8

Ans D it is!

Vips im totally lost in this... can u explain!!!
how u started g1 with even? based on answer choices?
if so how come u calculated g2?

If Result is even then two possibilities for argument = 1 Even 1 Odd
If Result is odd then one possibility for argument = 1 Even

Anyway, to start from scratch:
how u started g1 with even? based on answer choices?
question says,
g(x) = x/2 , if x is even=> Observation: if x is even, result is even/2 which could be odd or even.
g(x) = x+5, if x is odd => Observation: if x is odd, result is always even. (odd number+5= even number)

Another way to get there :

We know final result is 19. that is:
g(something) =19
Now what is this something? it could be 38 giving 19 when divided by 2. Or it could be 14 when 5 is added.
However, it can not be 14 because 14 is even and g(14) will be 7 not 19 by the definition of g(x). So there is only possiblity 38.
So if result is odd, then argument must have been even.

Therefore for argument of g1, you start with Even since the result is odd (19).

if so how come u calculated g2
Lets again see, we found out that argument of g1 was even. Now this even could have been result of another even number or an odd number. Let see the example:
taking forward previous values. We found above that argument for g1 is 38.
now, argument for g2? we know that g2(something) =38
What is this something? it could be 76, which gives 38 when divided by 2. Or it could be 33 which gives 38 when 5 is added. Both of these values are possible as per g(x) definition.

It can not be a gmat question. but its good fun.

to summarize, try to understand these lines:
If Result is even then two possibilities for argument = 1 Even 1 Odd
If Result is odd then one possibility for argument = 1 Even
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Re: The function g(x) is defined for integers x such that if x [#permalink]  16 Jan 2013, 05:36
12
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Expert's post
MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?

A. 1
B. 5
C. 7
D. 8
E. 11

Notice that when x is odd, g(x) = x + 5 (Recall that Odd + Odd = Even)
This means g(x) becomes even when x is odd. So if g(x) is odd, x MUST have been even.

Since g(g(g(g(g(x))))) = 19, we can say that g(g(g(g(x)))) must be even i.e. 19*2 = 38

Since g(g(g(g(x)))) = 38, g(g(g(x))) can be either even or odd so it can take 2 values: 38*2 = 76 or 38 - 5 = 33

If g(g(g(x))) = 76 g(g(x)) can again take two values - one even and one odd
If g(g(g(x))) = 33, g(g(x)) MUST be even 33*2 = 66.

So g(g(x)) can take 3 values: 2 even and one odd.

Notice that every even value gives you 2 values of the inner expression - one even and one odd - and every odd value gives you only one even value of the inner expression.

Then g(x) can take 5 different values - 3 even and 2 odd
Then x can take 8 different values - 5 even and 3 odd

An example of pattern recognition.
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Re: The function g(x) is defined for integers x such that if x [#permalink]  11 Aug 2013, 04:18
VeritasPrepKarishma wrote:
MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?

A. 1
B. 5
C. 7
D. 8
E. 11

Notice that when x is odd, g(x) = x + 5 (Recall that Odd + Odd = Even)
This means g(x) becomes even when x is odd. So if g(x) is odd, x MUST have been even.

Since g(g(g(g(g(x))))) = 19, we can say that g(g(g(g(x)))) must be even i.e. 19*2 = 38

Since g(g(g(g(x)))) = 38, g(g(g(x))) can be either even or odd so it can take 2 values: 38*2 = 76 or 38 - 5 = 33

If g(g(g(x))) = 76 g(g(x)) can again take two values - one even and one odd
If g(g(g(x))) = 33, g(g(x)) MUST be even 33*2 = 66.

So g(g(x)) can take 3 values: 2 even and one odd.

Notice that every even value gives you 2 values of the inner expression - one even and one odd - and every odd value gives you only one even value of the inner expression.

Then g(x) can take 5 different values - 3 even and 2 odd
Then x can take 8 different values - 5 even and 3 odd

An example of pattern recognition.

Thank you..Now it's clear for me..
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Re: The function g(x) is defined for integers x such that if x [#permalink]  11 Sep 2013, 19:44
VeritasPrepKarishma wrote:
MOKSH wrote:
The function g(x) is defined for integers x such that if x is even, g(x) = x/2 and if x is odd, g(x) = x + 5. Given that g(g(g(g(g(x))))) = 19, how many possible values for x would satisfy this equation?

A. 1
B. 5
C. 7
D. 8
E. 11

Notice that when x is odd, g(x) = x + 5 (Recall that Odd + Odd = Even)
This means g(x) becomes even when x is odd. So if g(x) is odd, x MUST have been even.

Since g(g(g(g(g(x))))) = 19, we can say that g(g(g(g(x)))) must be even i.e. 19*2 = 38

Since g(g(g(g(x)))) = 38, g(g(g(x))) can be either even or odd so it can take 2 values: 38*2 = 76 or 38 - 5 = 33

If g(g(g(x))) = 76 g(g(x)) can again take two values - one even and one odd
If g(g(g(x))) = 33, g(g(x)) MUST be even 33*2 = 66.

So g(g(x)) can take 3 values: 2 even and one odd.

Notice that every even value gives you 2 values of the inner expression - one even and one odd - and every odd value gives you only one even value of the inner expression.

Then g(x) can take 5 different values - 3 even and 2 odd
Then x can take 8 different values - 5 even and 3 odd

An example of pattern recognition.

Excellent, You write the most amazing solutions here.
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Re: The function g(x) is defined for integers x such that if x [#permalink]  03 Mar 2014, 00:38
this seems to be one of an easier questions from the list.
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Re: The function g(x) is defined for integers x such that if x   [#permalink] 03 Mar 2014, 00:38
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