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Math Expert V
Joined: 02 Sep 2009
Posts: 59590

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3 00:00

Difficulty:   55% (hard)

Question Stats: 50% (01:22) correct 50% (01:28) wrong based on 161 sessions

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If $$f(x) = \frac{x}{x + 1}$$, what is $$f(\frac{1}{x})$$ in terms of $$f(x)$$?

A. $$f(x)$$
B. $$-f(x)$$
C. $$\frac{1}{f(x)}$$
D. $$1 - f(x)$$
E. none of the above

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Math Expert V
Joined: 02 Sep 2009
Posts: 59590

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Official Solution:

If $$f(x) = \frac{x}{x + 1}$$, what is $$f(\frac{1}{x})$$ in terms of $$f(x)$$?

A. $$f(x)$$
B. $$-f(x)$$
C. $$\frac{1}{f(x)}$$
D. $$1 - f(x)$$
E. none of the above

$$f(\frac{1}{x})= \frac{\frac{1}{x}}{\frac{1}{x} + 1} = \frac{1}{1 + x} = \frac{1 + x - x}{1 + x} = 1 - \frac{x}{1 + x} = 1 - f(x)$$.

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Manager  Joined: 08 Feb 2014
Posts: 200
Location: United States
Concentration: Finance
GMAT 1: 650 Q39 V41 WE: Analyst (Commercial Banking)

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Hi,

How did you get from 1/(1+x) to (1+x-x)/(1+x), and then from this to the final step?

Thanks
Intern  Joined: 30 Jul 2014
Posts: 4
GMAT 1: 540 Q40 V25 GMAT 2: 580 Q47 V24 ### Show Tags

9
JackSparr0w

To solve this question,

First find the value of f(1/x). To do this, replace x with 1/x in the equation.

so f(1/x)=(1/x)/((1/x)+1) = 1/1+x

Now check which option will give you 1/1+x.

Option A : f(x) = x/x+1 not equal..

Option B : -f(x) = -x/x+1 not equal..

Option C : 1/f(x) = x+1/x not equal..

Option D : 1-f(x) = 1-(x/x+1) =(x+1-x)/x+1 = 1/x+1 .. This is the answer.

Option E : None..

+1 Kudos if this helps..
Math Expert V
Joined: 02 Sep 2009
Posts: 59590

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JackSparr0w wrote:
Hi,

How did you get from 1/(1+x) to (1+x-x)/(1+x), and then from this to the final step?

Thanks

$$f(\frac{1}{x})= \frac{\frac{1}{x}}{\frac{1}{x} + 1} =$$.

$$\frac{\frac{1}{x}}{(\frac{1+x}{x})} =$$

$$\frac{1}{x}*\frac{x}{x+1}$$

$$=\frac{1}{1 + x} =$$

$$=\frac{(1 + x) - x}{1 + x} =$$

$$=\frac{(1 + x)}{1 + x}-\frac{x}{1+x}=$$

$$=1 - \frac{x}{1 + x} =$$

$$=1 - f(x)$$
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Manager  Joined: 08 Feb 2014
Posts: 200
Location: United States
Concentration: Finance
GMAT 1: 650 Q39 V41 WE: Analyst (Commercial Banking)

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2
msaid007 wrote:
JackSparr0w

To solve this question,

First find the value of f(1/x). To do this, replace x with 1/x in the equation.

so f(1/x)=(1/x)/((1/x)+1) = 1/1+x

Now check which option will give you 1/1+x.

Option A : f(x) = x/x+1 not equal..

Option B : -f(x) = -x/x+1 not equal..

Option C : 1/f(x) = x+1/x not equal..

Option D : 1-f(x) = 1-(x/x+1) =(x+1-x)/x+1 = 1/x+1 .. This is the answer.

Option E : None..

+1 Kudos if this helps..

Thanks! Got it!

For others, I think it may be helpful to elaborate on one other point. After you find that f(1/x) = 1/(1+x), compare this to the answer choices by substituting a value for x.

Ex: (assume x = 2). When x is 2, 1/(1+x) becomes 1/(1+2) or (1/3).

We also know f(x) = x/(x+1) from the question stem. So when x=2, f(x)=2/(2+1), or 2/3. We need to find the answer choice that changes 2/3 to 1/3.

Knowing this we can see that choice D is correct. Choice D is 1-f(x), or 1-(2/3) which equals 1/3; the same as our answer above.
Manager  Joined: 08 Feb 2014
Posts: 200
Location: United States
Concentration: Finance
GMAT 1: 650 Q39 V41 WE: Analyst (Commercial Banking)

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Bunuel wrote:
JackSparr0w wrote:
Hi,

How did you get from 1/(1+x) to (1+x-x)/(1+x), and then from this to the final step?

Thanks

$$f(\frac{1}{x})= \frac{\frac{1}{x}}{\frac{1}{x} + 1} =$$.

$$\frac{\frac{1}{x}}{(\frac{1+x}{x})} =$$

$$\frac{1}{x}*\frac{x}{x+1}$$

$$=\frac{1}{1 + x} =$$

$$=\frac{(1 + x) - x}{1 + x} =$$

$$=\frac{(1 + x)}{1 + x}-\frac{x}{1+x}=$$

$$=1 - \frac{x}{1 + x} =$$

$$=1 - f(x)$$

Thanks Bunuel, much appreciated!. I think I'll have to use the substitution approach here, though.
Manager  Joined: 24 Nov 2013
Posts: 55

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Bunuel

I was able to get f(1/x)= 1/1+x.
this is nothing but (x)/(x)(1+x)

and we know f(x) = x/x+1
therefore f(1/x) = f(x) divided by x
but this is not available in the options. Hence I chose E.

Could you please tell me what is wrong with this approach. Thanks.
Manager  G
Joined: 02 Jun 2015
Posts: 169
Location: Ghana

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Bunuel wrote:
If $$f(x) = \frac{x}{x + 1}$$, what is $$f(\frac{1}{x})$$ in terms of $$f(x)$$?

A. $$f(x)$$
B. $$-f(x)$$
C. $$\frac{1}{f(x)}$$
D. $$1 - f(x)$$
E. none of the above
[quote="Bunuel"]If $$f(x) = \frac{x}{x + 1}$$, what is $$f(\frac{1}{x})$$ in terms of $$f(x)$$?

f(x) =x/(x+1) let's x = 2, then f(x) = 2/(2+1) = 2/3 and f(1/x) = (1/2)/(3/2) = 1/3

Now f(x) = 2/3, plug in 2/3 into the answer choices starting with choice D. Even before you start plugging in, you realise that D is the only answer choice that will result in 1/3 = f(1/x)
Intern  Joined: 18 May 2016
Posts: 6

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Bunuel wrote:
JackSparr0w wrote:
Hi,

How did you get from 1/(1+x) to (1+x-x)/(1+x), and then from this to the final step?

Thanks

$$f(\frac{1}{x})= \frac{\frac{1}{x}}{\frac{1}{x} + 1} =$$.

$$\frac{\frac{1}{x}}{(\frac{1+x}{x})} =$$

$$\frac{1}{x}*\frac{x}{x+1}$$

$$=\frac{1}{1 + x} =$$

$$=\frac{(1 + x) - x}{1 + x} =$$

$$=\frac{(1 + x)}{1 + x}-\frac{x}{1+x}=$$

$$=1 - \frac{x}{1 + x} =$$

$$=1 - f(x)$$

Can you explain the reasoning behind the highlighted steps please
Math Expert V
Joined: 02 Sep 2009
Posts: 59590

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bjklue wrote:
Bunuel wrote:
JackSparr0w wrote:
Hi,

How did you get from 1/(1+x) to (1+x-x)/(1+x), and then from this to the final step?

Thanks

$$f(\frac{1}{x})= \frac{\frac{1}{x}}{\frac{1}{x} + 1} =$$.

$$\frac{\frac{1}{x}}{(\frac{1+x}{x})} =$$

$$\frac{1}{x}*\frac{x}{x+1}$$

$$=\frac{1}{1 + x} =$$

$$=\frac{(1 + x) - x}{1 + x} =$$

$$=\frac{(1 + x)}{1 + x}-\frac{x}{1+x}=$$

$$=1 - \frac{x}{1 + x} =$$

$$=1 - f(x)$$

Can you explain the reasoning behind the highlighted steps please

By isolating x/(1+x) we are able to get f(x) in the expression.
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Intern  B
Joined: 30 Aug 2016
Posts: 20
GMAT 1: 490 Q36 V22 GPA: 3

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I have reached the substitution but unable to get the process after that...
Intern  B
Joined: 02 Jun 2015
Posts: 20
Location: United States

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Tanyanagra wrote:
I have reached the substitution but unable to get the process after that...

Hi Tanyanagra,

For me using substitution is better than trying to manipulate the variables, especially under pressure....here's how I solve it !

Substitute x=2 into this 1st equation: f(x) = x divided by x+1 f(2) = 2 divided by 2+1 = 2/3

Then, the question asks what is f(1/x) in terms of f(x)?

You'll need to convert the f(x) to f(1/x) that will be 1/x divided by 1/x+1

Substitute x=2 into this 2nd equation f(1/x): 1/2 divided by 1/2+1 = 1/2 divided by 3/2 = 1/2 * 2/3 = 1/3

Now, you have to find 1/3in the answer choices:

A) f(x) = 2/3 not a match

B) - f(x) = - 2/3 not a match

C) 1/f(x) = 1 divided by 2/3 = 3/2 not a match

D) 1 - f(x) = 1 - 2/3 = 1/3 BINGO!! That's a match

E) none of the above = wrong

I hope it helps!

Thanks Ale! Manager  S
Joined: 26 Sep 2018
Posts: 56
Location: Sweden

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In problems like these, one of the best ways to solve is by plugging in numbers.

For example:

x=5 and 1/x = 1/5.

So f(5)=5/6 and f(1/5)= 1/6, only option d works.

Try x= 1/3 to confirm. Admittedly not the best method, but it does the trick!
Manager  G
Joined: 22 Jun 2017
Posts: 166
Location: Argentina
Schools: HBS, Stanford, Wharton
GMAT 1: 630 Q43 V34 ### Show Tags

I think this is a 700 question.
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Intern  B
Joined: 16 Feb 2019
Posts: 15

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Answer should only be in terms of f(x), else f(1/x) = f(x)/ x Re M19-14   [#permalink] 19 Sep 2019, 23:29
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# M19-14

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