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Math Revolution GMAT Instructor V
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f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 55% (02:07) correct 45% (02:25) wrong based on 38 sessions

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[GMAT math practice question]

$$f(2x-1) = \frac{( x + 2 )}{( x – 2 )}$$. What is $$f(x)$$?

$$A. f(x) = \frac{( x + 5 )}{( x – 3 )}$$
$$B. f(x) = \frac{( x - 5 )}{( x + 3 )}$$
$$C. f(x) = \frac{( x + 5 )}{( x + 3 )}$$
$$D. f(x) = \frac{( x – 5 )}{( x – 3 )}$$
$$E. f(x) = \frac{( x + 3 )}{( x – 3 )}$$

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Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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f(2x-1) = f(x)
Now we have to replace 2x-1 by x, i.e, 2x-1 = x
then x = 1
f(2x-1) = -3

Now put x = 1 in all the options, Only Option A yields -3.

A is the answer.
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Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

$$f(2x-1) = \frac{( x + 2 )}{( x – 2 )}$$. What is $$f(x)$$?

$$A. f(x) = \frac{( x + 5 )}{( x – 3 )}$$
$$B. f(x) = \frac{( x - 5 )}{( x + 3 )}$$
$$C. f(x) = \frac{( x + 5 )}{( x + 3 )}$$
$$D. f(x) = \frac{( x – 5 )}{( x – 3 )}$$
$$E. f(x) = \frac{( x + 3 )}{( x – 3 )}$$

from given answer options
do plugin of f(x) as f(2x-1)
in option A
we get
$$f(2x-1) = \frac{( x + 2 )}{( x – 2 )}$$

IMO A
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Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

$$f(2x-1) = \frac{( x + 2 )}{( x – 2 )}$$. What is $$f(x)$$?

$$A. f(x) = \frac{( x + 5 )}{( x – 3 )}$$
$$B. f(x) = \frac{( x - 5 )}{( x + 3 )}$$
$$C. f(x) = \frac{( x + 5 )}{( x + 3 )}$$
$$D. f(x) = \frac{( x – 5 )}{( x – 3 )}$$
$$E. f(x) = \frac{( x + 3 )}{( x – 3 )}$$

Let x=1.
Plugging x=1 into $$f(2x-1) = \frac{x + 2}{x – 2}$$, we get:
$$f(2*1-1) = \frac{1 + 2}{1 – 2}$$
$$f(1) = -3$$

When x=1, the question stem becomes:
What is $$f(1)$$?
Since f(1) = -3, the correct answer must yield -3 when x=1.
Only A works:
$$f(1) = \frac{1 + 5}{1-3}$$ = -3

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Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

$$f(2x-1) = \frac{( x + 2 )}{( x – 2 )}$$. What is $$f(x)$$?

$$A. f(x) = \frac{( x + 5 )}{( x – 3 )}$$
$$B. f(x) = \frac{( x - 5 )}{( x + 3 )}$$
$$C. f(x) = \frac{( x + 5 )}{( x + 3 )}$$
$$D. f(x) = \frac{( x – 5 )}{( x – 3 )}$$
$$E. f(x) = \frac{( x + 3 )}{( x – 3 )}$$

$$? = f\left( x \right)$$

$$f\left( {2x - 1} \right) = {{x + 2} \over {x - 2}}\,\,\,\,\,\left( * \right)$$

$$2x - 1 = y\,\,\,\,\, \Rightarrow \,\,\,\,x = {{y + 1} \over 2}$$

$$\left( * \right)\,\,\,\, \Rightarrow \,\,\,\,f\left( y \right)\,\, = \,\,{{{{y + 1} \over 2} + 2} \over {{{y + 1} \over 2} - 2}}\,\, = \,\,{{y + 1 + 4} \over {y + 1 - 4}} = {{y + 5} \over {y - 3}}$$

$$?\,\,\,:\,\,\,f\left( y \right) = {{y + 5} \over {y - 3}}\,\,\,\,\,\,\,\left[ {\,f\left( x \right) = {{x + 5} \over {x - 3}}\,\,\,{\rm{if}}\,\,{\rm{you}}\,\,{\rm{prefer}}!\,} \right]$$

The correct answer is therefore (A).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Math Revolution GMAT Instructor V
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Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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=>

We solve this problem using a change of variable. Let $$t = 2x -1.$$
Then $$2x = t + 1$$ or $$x = \frac{( t + 1 )}{2.}$$
$$f(2x-1) = f(t) = ( (t+1)/2 + 2 ) / ( (t+1)/2 – 2 ) = ( t + 1 + 4 ) / ( t + 1 – 4 )$$(after multiplying both top and bottom by 2)
$$= \frac{( t + 5 )}{( t – 3 ).}$$
Making the substitution $$t = x$$ gives
$$f(x) = \frac{( x + 5 )}{( x – 3 )}.$$

Therefore, the answer is A.
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Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

$$f(2x-1) = \frac{( x + 2 )}{( x – 2 )}$$. What is $$f(x)$$?

$$A. f(x) = \frac{( x + 5 )}{( x – 3 )}$$
$$B. f(x) = \frac{( x - 5 )}{( x + 3 )}$$
$$C. f(x) = \frac{( x + 5 )}{( x + 3 )}$$
$$D. f(x) = \frac{( x – 5 )}{( x – 3 )}$$
$$E. f(x) = \frac{( x + 3 )}{( x – 3 )}$$

Let y = 2x - 1, so x = (y + 1)/2. In other words,

f(y) = [(y + 1)/2 + 2] / [(y + 1)/2 - 2]

f(y) = (y + 1 + 4) / (y + 1 - 4)

f(y) = (y + 5) / (y - 3)

Now we can replace y with x and obtain:

f(x) = (x + 5) / (x - 3)

Alternate Solution:

First, let’s take x = -2 in f(2x - 1) = (x + 2) / (x - 2):

f( 2(-2) - 1) = (-2 + 2) / (-2 - 2)

f( -5) = 0

Looking at the answer choices, we observe that only A and C equal zero when x = -5; therefore we eliminate B, D and E.

Next, let’s take x = 3 in f(2x - 1) = (x + 2) / (x - 2):

f( 2(3) - 1) = (3 + 2) / (3 - 2)

f(5) = 5

In answer choice C, when we take x = 5, we obtain f(5) = 10/8, which is not equal to 5. Therefore, we eliminate C as well. The only remaining answer choice is A. Indeed, for the function in A, if we take x = 5, we obtain f(5) = (5 + 5) / (5 - 3) = 10/2 = 5.

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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?   [#permalink] 06 Feb 2019, 19:34
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