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The function f is defined for all nonzero x by the equation f(x) = x -

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The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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Updated on: 21 Nov 2016, 21:12
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The function f is defined for all nonzero x by the equation $$f(x) = x - \frac{1}{x}$$. If $$x\neq{0}$$, which of the following equals $$f(\frac{1}{x})$$?

A. $$f(x)$$

B. $$f(-x)$$

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

D. $$\frac{1}{f(x)}$$

E. $$-\frac{1}{f(x)}$$
[Reveal] Spoiler: OA

Originally posted by felippemed on 21 Nov 2016, 14:28.
Last edited by Bunuel on 21 Nov 2016, 21:12, edited 1 time in total.
Edited the question.
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Posts: 4670
Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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21 Nov 2016, 16:54
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felippemed wrote:
the function $$f$$ is defined for all nonzero x by the equation $$f(x) = x - \frac{1}{x}$$. If $$x\neq{0}$$, which of the following equals $$f(\frac{1}{x})$$?

A.$$f(x)$$
B. $$f(-x)$$
C. $$f(\frac{-1}{x})$$
D. $$\frac{1}{f(x)}$$
E. $$\frac{-1}{f(x)}$$

Dear felippemed,

This is a great question. I'm happy to respond.

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

When we plug 1/x into the argument of the function, the first time, x, becomes simply 1/x. The second term, 1/x, becomes 1/(1/x) = x. Thus,

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

Notice that -f(x) would be a valid choice, but that's not an answer. As it happens, when we simply put a negative sign, -x, into the argument, each term becomes negative, so in this instance, -f(x) = f(-x). (In Precalculus, this would tell us that the function's graph has odd symmetry, but this is beyond what you need to know for the GMAT).

Does all this make sense?
Mike
_________________

Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Joined: 23 Jun 2009
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GMAT 1: 470 Q30 V20
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Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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21 Nov 2016, 17:17
Quote:
Does all this make sense? Mike

No hahaha

If there is an example plugging number, it could clarify a little more.

Thanks anyway
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Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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17 Apr 2017, 19:55
mikemcgarry wrote:
felippemed wrote:
the function $$f$$ is defined for all nonzero x by the equation $$f(x) = x - \frac{1}{x}$$. If $$x\neq{0}$$, which of the following equals $$f(\frac{1}{x})$$?

A.$$f(x)$$
B. $$f(-x)$$
C. $$f(\frac{-1}{x})$$
D. $$\frac{1}{f(x)}$$
E. $$\frac{-1}{f(x)}$$

Dear felippemed,

This is a great question. I'm happy to respond.

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

When we plug 1/x into the argument of the function, the first time, x, becomes simply 1/x. The second term, 1/x, becomes 1/(1/x) = x. Thus,

$$f(\frac{1}{x}) = \frac{1}{x} - x = -(x - \frac{1}{x}) = -f(x)$$
Notice that -f(x) would be a valid choice, but that's not an answer. As it happens, when we simply put a negative sign, -x, into the argument, each term becomes negative, so in this instance, -f(x) = f(-x). (In Precalculus, this would tell us that the function's graph has odd symmetry, but this is beyond what you need to know for the GMAT).

Does all this make sense?
Mike

Hi Mike, I determined f(1/x) as (1/x)-x and then worked through the functions in answer choices, landing at B. Do you see any pitfalls to this approach?
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Posts: 4670
Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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23 May 2017, 16:43
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Cez005 wrote:
Hi Mike, I determined f(1/x) as (1/x)-x and then worked through the functions in answer choices, landing at B. Do you see any pitfalls to this approach?

Dear Cez005,

I'm happy to respond.

You ask if there were any pitfalls. No and yes. I am sure your algebra was superb. The drawback is on a larger scale.

To discuss this, I will introduce the following distinction.

Left brain thinking = rule-based; good at step-by-step recipes and procedures; operates with logic and analysis; proceeds step-by-step

Right brain thinking = pattern-based; good at seeing complex connections and larger patterns; operates with analogy and association; proceeds by non-linear leaps

You can read more on this blog:
How to do GMAT Math Faster

You see, left-brain thinkers love to algebra, and they look for any opportunity to use algebra in a step-by-step solution. Even if all the algebra is flawless, the problem with this approach is that it often takes too long. In fact, the GMAT Quant, on higher level questions, loves to create question that are complete traps for someone who opts for the straightforward algebraic solution. This GMAT Prep problem is along these lines.

When I looked at the problem, I solved it in my head in under 10 seconds by observing the patterns. I tried to make this clear in my explanation.

I don't know you, so I don't know whether you are predominately a left-brain thinker. I will say the solution method you described was an extreme left-brain lengthy solution for a problem that can be completed quite quickly with a right-brain approach.

This is the paradox of growth. You are not really preparing for the GMAT if you keep doing the things that already come naturally to you. You improve by stretching yourself to get at least a little better in the areas that are completely unfamiliar.

Does all this make sense?
Mike
_________________

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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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03 Jun 2017, 14:50
Bump

Reading the method of plugging in makes a lot of sense and I understand that. I am having an issue going the algebraic route.

Can anyone explain why D is incorrect algebraically? (I sense this is a brain fart on my end but I can't figure it out)
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4670
The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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05 Jun 2017, 11:38
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gmathopeful19 wrote:
Bump

Reading the method of plugging in makes a lot of sense and I understand that. I am having an issue going the algebraic route.

Can anyone explain why D is incorrect algebraically? (I sense this is a brain fart on my end but I can't figure it out)

Dear gmathopeful19,

I'm happy to respond.

My friend, I believe you have fallen into what is known as a robust mistake. In pedagogical research, a robust mistake is one to which students return even after they have had a moment of fully understanding why it is wrong. Here is this particular mistake pattern:

$$\dfrac{1}{A + B} \neq \dfrac{1}{A} + \dfrac{1}{B}$$

Others include

$$(A \pm B)^2 \neq A^2 \pm B^2$$

$$\sqrt{A \pm B} \neq \sqrt{A} \pm \sqrt{B}$$

All of these are incorrect overgeneralizations of the Distributive Law. The Distributive Law says that multiplication & division distribute over addition & subtraction.

$$C \times (A \pm B) = C \times A \pm C \times B$$

$$\dfrac{A \pm B}{C} = \dfrac{A}{C} \pm \dfrac{A}{C}$$

Multiplication & division distribute, but basically, nothing else distributes. All of these mistake are incorrect extensions of the pattern of the Distributive Law to things that don't distribute over addition & subtraction. Be very careful: even when you fully understand why the above mistakes are in fact mistakes, when you are tired or under stress, you mind almost on automatic pilot will think the mistake pattern is correct. It takes quite a bit of effort to root out a robust mistake.

Does all this make sense?
Mike
_________________

Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Manager
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Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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05 Jun 2017, 12:16
mikemcgarry wrote:
gmathopeful19 wrote:
Bump

Reading the method of plugging in makes a lot of sense and I understand that. I am having an issue going the algebraic route.

Can anyone explain why D is incorrect algebraically? (I sense this is a brain fart on my end but I can't figure it out)

Dear gmathopeful19,

I'm happy to respond.

My friend, I believe you have fallen into what is known as a robust mistake. In pedagogical research, a robust mistake is one to which students return even after they have had a moment of fully understanding why it is wrong. Here is this particular mistake pattern:

$$\dfrac{1}{A + B} \neq \dfrac{1}{A} + \dfrac{1}{B}$$

Others include

$$(A \pm B)^2 \neq A^2 \pm B^2$$

$$\sqrt{A \pm B} \neq \sqrt{A} \pm \sqrt{B}$$

All of these are incorrect overgeneralizations of the Distributive Law. The Distributive Law says that multiplication & division distribute over addition & subtraction.

$$C \times (A \pm B) = C \times A \pm C \times B$$

$$\dfrac{A \pm B}{C} = \dfrac{A}{C} \pm \dfrac{A}{C}$$

Multiplication & division distribute, but basically, nothing else distributes. All of these mistake are incorrect extensions of the pattern of the Distributive Law to things that don't distribute over addition & subtraction. Be very careful: even when you fully understand why the above mistakes are in fact mistakes, when you are tired or under stress, you mind almost on automatic pilot will think the mistake pattern is correct. It takes quite a bit of effort to root out a robust mistake.

Does all this make sense?
Mike

Yes I definitely know those rules ha. Long day I guess. Appreciate the help

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The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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Updated on: 13 Feb 2018, 18:28
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Hi All,

This question can be solved by TESTing VALUES.

We're told that f(X) = X - (1/X) and that X does not equal 0. We're asked for the value of f(1/X).

IF....
X = 2
f(1/2) = 1/2 - (1/(.5) = 1/2 - 2 = -3/2

So we're looking for an answer that equals -3/2 when we plug X=2 into the answers...

Answer A: f(2) = 2 - 1/2 = 3/2 NOT a match
Answer B: f(-2) = -2 - (1/-2) = -3/2 This IS a match
Answer C: f(-1/2) = -.5 - (1/-.5) = +3/2 NOT a match

We can use the calculation from Answer A to deal with Answers D and E....
Answer D: 1/f(2) = 1/(3/2) = 2/3 NOT a match
Answer E: -1/f(2) = -1/(3/2) = -2/3 NOT a match

[Reveal] Spoiler:
B

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

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Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Originally posted by EMPOWERgmatRichC on 28 Jan 2018, 15:27. Last edited by EMPOWERgmatRichC on 13 Feb 2018, 18:28, edited 1 time in total. Intern Joined: 10 Sep 2017 Posts: 6 Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink] Show Tags 05 Feb 2018, 11:24 EMPOWERgmatRichC wrote: Hi All, This question can be solved by TESTing VALUES. We're told that f(X) = X - (1/X) and that X does not equal 0. We're asked for the value of f(1/X). IF.... X = 2 f(1/2) = 1/2 - (1/(.5) = 1/2 - 2 = -3/2 So we're looking for an answer that equals -3/2 when we plug X=2 into the answers... Answer A: f(2) = 2 - 1/2 = 3/2 NOT a match Answer B: f(-2) = -2 - (1/-2) = -3/2 This IS a match Answer C: f(-1/2) = -.5 - (1/-.5) = -5/2 NOT a match We can use the calculation from Answer A to deal with Answers D and E.... Answer D: 1/f(2) = 1/(3/2) = 2/3 NOT a match Answer E: -1/f(2) = -1/(3/2) = -2/3 NOT a match Final Answer: [Reveal] Spoiler: B GMAT assassins aren't born, they're made, Rich Why did you put (.5) for x: (1/x) ? EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 11633 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink] Show Tags 05 Feb 2018, 11:52 Hi cman2010, This question asks us for the value of f(1/X). If we TEST X=2, then we have to plug 2 wherever an "X" appears. To start, that's in the value of 1/X (which would make that value 1/2), which we then plug into the given function. GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

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Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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12 Feb 2018, 21:21
EMPOWERgmatRichC wrote:
Hi All,

This question can be solved by TESTing VALUES.

We're told that f(X) = X - (1/X) and that X does not equal 0. We're asked for the value of f(1/X).

IF....
X = 2
f(1/2) = 1/2 - (1/(.5) = 1/2 - 2 = -3/2

So we're looking for an answer that equals -3/2 when we plug X=2 into the answers...

Answer A: f(2) = 2 - 1/2 = 3/2 NOT a match
Answer B: f(-2) = -2 - (1/-2) = -3/2 This IS a match
Answer C: f(-1/2) = -.5 - (1/-.5) = -5/2 NOT a match

We can use the calculation from Answer A to deal with Answers D and E....
Answer D: 1/f(2) = 1/(3/2) = 2/3 NOT a match
Answer E: -1/f(2) = -1/(3/2) = -2/3 NOT a match

[Reveal] Spoiler:
B

GMAT assassins aren't born, they're made,
Rich

I got -3/2 for both B and C...

Answer C: f(-1/2) = -1/2 - (1/-1/2) = -3/2

What did I do wrong?
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Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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13 Feb 2018, 18:28
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Hi OCDianaOC,

It looks like we both made errors in our calculations (I've edited my explanation accordingly).

With Answer C, we're dealing with....
-0.5 - (1/-0.5) =
-1/2 - (-2) =
-1/2 + 2 =
-1/2 + 4/2 =
+3/2

That's NOT a match for what we're looking for (we're looking for an answer that equals -3/2).

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Intern
Joined: 16 Oct 2017
Posts: 37
Re: The function f is defined for all nonzero x by the equation f(x) = x - [#permalink]

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13 Feb 2018, 19:50
EMPOWERgmatRichC wrote:
Hi OCDianaOC,

It looks like we both made errors in our calculations (I've edited my explanation accordingly).

With Answer C, we're dealing with....
-0.5 - (1/-0.5) =
-1/2 - (-2) =
-1/2 + 2 =
-1/2 + 4/2 =
+3/2

That's NOT a match for what we're looking for (we're looking for an answer that equals -3/2).

GMAT assassins aren't born, they're made,
Rich

Oh, you're right! Okay, I get it now

Thanks for showing this method!
Re: The function f is defined for all nonzero x by the equation f(x) = x -   [#permalink] 13 Feb 2018, 19:50
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