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M19-14

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Math Expert
Joined: 02 Sep 2009
Posts: 58116

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16 Sep 2014, 01:06
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Difficulty:

55% (hard)

Question Stats:

50% (01:23) correct 50% (01:27) wrong based on 158 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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28 Oct 2014, 05:05
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
Joined: 02 Sep 2009
Posts: 58116

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28 Oct 2014, 05:09
2
1
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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03 Nov 2014, 21:06
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.
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Posts: 58116

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16 Sep 2014, 01:06
1
1
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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27 Oct 2014, 19:21
1
Hi,

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

Thanks
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03 Nov 2014, 21:08
1
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.
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12 Oct 2015, 18:38
1
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.
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15 Aug 2016, 07:50
1
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)
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15 Dec 2016, 01:06
1
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
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Posts: 58116

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15 Dec 2016, 03:37
1
1
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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16 Jan 2017, 22:43
1
I have reached the substitution but unable to get the process after that...
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30 May 2018, 15:51
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!
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30 Dec 2018, 12:53
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!
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13 Feb 2019, 08:56
I think this is a 700 question.
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19 Sep 2019, 23:29
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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