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f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?

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f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?

by Max@Math Revolution » Thu Jan 31, 2019 12:59 am

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[GMAT math practice question]

f(2x-1) = ( x + 2 ) / ( x - 2 ). What is f(x)?

A. f(x) = ( x + 5 ) / ( x - 3 )
B. f(x) = ( x - 5 ) / ( x + 3 )
C. f(x) = ( x + 5 ) / ( x + 3 )
D. f(x) = ( x - 5 ) / ( x - 3 )
E. f(x) = ( x + 3 ) / ( x - 3 )
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by GMATGuruNY » Thu Jan 31, 2019 3:28 am
Max@Math Revolution wrote:[GMAT math practice question]

f(2x-1) = ( x + 2 ) / ( x - 2 ). What is f(x)?

A. f(x) = ( x + 5 ) / ( x - 3 )
B. f(x) = ( x - 5 ) / ( x + 3 )
C. f(x) = ( x + 5 ) / ( x + 3 )
D. f(x) = ( x - 5 ) / ( x - 3 )
E. f(x) = ( x + 3 ) / ( x - 3 )
Let x=1.
Plugging x=1 into f(2x-1) = ( x + 2 ) / ( x - 2 ), we get:
f(2*1 - 1) = (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) = (1+5)/(1-3) = 6/-2 = -3.

The correct answer is A.
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f(2x-1) = ( x + 2 ) / ( x – 2 ). What is f(x)?

by fskilnik@GMATH » Thu Jan 31, 2019 4:23 am
Max@Math Revolution wrote:[GMAT math practice question]

f(2x-1) = ( x + 2 ) / ( x - 2 ). What is f(x)?

A. f(x) = ( x + 5 ) / ( x - 3 )
B. f(x) = ( x - 5 ) / ( x + 3 )
C. f(x) = ( x + 5 ) / ( x + 3 )
D. f(x) = ( x - 5 ) / ( x - 3 )
E. f(x) = ( 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).


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by Max@Math Revolution » Mon Feb 04, 2019 4:16 am
=>

We solve this problem using a change of variable. Let t = 2x -1.
Then 2x = t + 1 or x = ( 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)
= ( t + 5 ) / ( t - 3 ).
Making the substitution t = x gives
f(x) = ( x + 5 ) / ( x - 3 ).

Therefore, the answer is A.
Answer: A
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by Scott@TargetTestPrep » Wed Feb 06, 2019 6:22 pm
Max@Math Revolution wrote:[GMAT math practice question]

f(2x-1) = ( x + 2 ) / ( x - 2 ). What is f(x)?

A. f(x) = ( x + 5 ) / ( x - 3 )
B. f(x) = ( x - 5 ) / ( x + 3 )
C. f(x) = ( x + 5 ) / ( x + 3 )
D. f(x) = ( x - 5 ) / ( x - 3 )
E. f(x) = ( 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.

Answer: A

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