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

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g(g(g(g(g(x))))) = 19

=> g(g(g(g(x)))) is even = 38 (if it were odd, the output is even)



now g(g(g(x))) can be 76 or 33

g(g(x)) can be 152, 71 or 66

g(x) can be 304, 147, 142, 132 or 61

x can be 608,299,294,284,137, 127, 264, 132

So total 8 values, hence the answer is D
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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

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Re: The function g(x) is defined for integers x such that if x [#permalink]

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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]

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New post 16 Jan 2013, 00:58
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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


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

Image

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Re: The function g(x) is defined for integers x such that if x [#permalink]

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New post 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]

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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?


ha ha.. the explanation was this:
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]

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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


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]

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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.


Karishma,

Your method is incorrect (though the answer is correct). You forgot that in such questions, there may be overlapping answers.
The only way to solve this is to find all possible values of x.
Solving step by step gives x = 122 or 127 or 264 or 294 or 608 or 299 or 284 or 137.
Hence (D).

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Re: The function g(x) is defined for integers x such that if x [#permalink]

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New post 12 Mar 2014, 02:38
RG800 wrote:
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.


Karishma,

Your method is incorrect (though the answer is correct). You forgot that in such questions, there may be overlapping answers.
The only way to solve this is to find all possible values of x.
Solving step by step gives x = 122 or 127 or 264 or 294 or 608 or 299 or 284 or 137.
Hence (D).


You are starting with 19 and performing 2 operations on it (*2 or -5) in different number and different order. Each chain of operations will give you a different result. You don't have to do it to find that out.
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Re: The function g(x) is defined for integers x such that if x [#permalink]

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New post 13 Mar 2014, 04:44
VeritasPrepKarishma wrote:
You are starting with 19 and performing 2 operations on it (*2 or -5) in different number and different order. Each chain of operations will give you a different result. You don't have to do it to find that out.



Why can't the *2 of one number be equal to the -5 of another? I don't get it :(

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Re: The function g(x) is defined for integers x such that if x [#permalink]

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New post 13 Mar 2014, 05:21
RG800 wrote:
VeritasPrepKarishma wrote:
You are starting with 19 and performing 2 operations on it (*2 or -5) in different number and different order. Each chain of operations will give you a different result. You don't have to do it to find that out.



Why can't the *2 of one number be equal to the -5 of another? I don't get it :(


You are starting with the same number 19.
Think about it: if you multiply 19 by 2 four times and subtract 5 once, can it be equal to if you multiply by 2 three times and subtract 5 twice?
Similarly, if you multiply by 2 four times and then subtract 5 once, can it be equal to if you subtract 5 once and then multiply by 2 four times.
The sequence in which operations are applied on a number change the number.
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Re: The function g(x) is defined for integers x such that if x [#permalink]

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New post 06 Aug 2014, 21:15
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!


I like your answer, but g1 should be g5, g2 should be g4, .... g5 should g1 in the possibilities section. Let me know If I am wrong
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Re: The function g(x) is defined for integers x such that if x [#permalink]

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New post 10 Nov 2016, 00:56
Best approach is to start with 19 and create a tree of possible values till 5 layers and eliminate options between even and odd.


Layer 1
19- 38 or 24

Layer 2
38 - 76 or 33
24 - 48 or 19

Layer 3
76 - 152 or 71
33 - 66 or 28
48 - 28 or 43
19 - 38 or 14(Eliminate 14 since it cannot be even number)

Layer 4 & Layer 5 keep on solving and eliminate based on even or odd number possibility. Final answer is 8 possible values

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