M19-14

Question

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.  1 + f(x)

Bunuel · Answer

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.  1 + f(x)

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)

Answer: D

JackSparr0w · Answer

Hi,

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

Thanks

msaid007 · Answer

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..

JackSparr0w · Answer

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.

Tanyanagra · Answer

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

Alecita · Answer

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 1/3 in 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!

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Rishikavats · Answer

------------------

Hi  Bunuel ,
It's not given to us that 'x' is not equal to 0 so how can we cancel 'x' when solving for f(1/x)??

Bunuel · Answer

If x were 0, 1/x would be undefined and f(1/x) wouldn't make any sense.

shaliny · Answer

hi Bunuel

after solving f(1/x)= 1/1+x, i multiplied and divide the R.H.S with x and got x/x(1+x) or 1/x *F(x), why is it wrong??

Bunuel · Answer

You went wrong for two reasons.

First, (1/x)*f(x) is not even among the answer choices.

Second, the question specifically asks to express f(1/x) in terms of f(x), but your expression still has both f(x) and x in it. That’s why it’s not valid.

shaliny · Answer

There was an option E- None , that's why i thought my calculation could be right:?
 according to second reason, i can't write as F(1/x) = f(x)/x, isn't? i mean this representation is not in option here but generally, is wrong or right as you said 1/x and f(x) both present .

Bunuel · Answer

1. Your algebraic manipulations are correct.
2. In terms of f(x) means there should be only f(x).
3. Modified option E.

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