Given f(x) = x/(x + 1), for what value k does f(f(k)) = 2/3 ?

Given f(x) = x/(x + 1)

so f(k)=k/k+1

f(f(k))=f(k/k+1)=f[(k/k+1)/ (k/k+1)+1)]

Given f(f(k))=2/3

f(k/k+1)=f[(k/k+1)/ (k/k+1)+1)]=2/3
By solving , we will get

k=-2

Answer is A

let's solve this problem by plugin approach.

let k=-2, then f(f(k))= f(f(-2))

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

hence answer should be A

Given f(x) =\(\frac{x}{(x + 1)}\), for what value k does f(f(k)) = \(\frac{2}{3}\) ?

A. –2
B.\(\frac{5}{3}\)
C. 1
D. 2
E. 8

Sol :

f(x) = \(\frac{x}{(x + 1)}\)
and
f(f(x) = \(\frac{x}{2x+1}\)........ 1

=> f(f(x) = \(\frac{2}{3}\) ..........2

equating 1 and 2

\(\frac{x}{2x+1}\) = \(\frac{2}{3}\)

Upon solving x = -2 hence ans A


Regards
SG

Starting with inner brackets :

f(f(k)) can be written as f(k/k+1)

Now, again applying the given condition on f(k/k+1)

So, f(k/k+1)= k/k+1/k/k+1 + 1 = 2/3

Or,k/2k+1 = 2/3

Or, k = -2

Ans : A

I too got A, -2 but with slightly different (and possibly convoluted) manner.

Lets say f(k) = y = x/(x+1). Solving backwards gives x=y/(1-y)

So if f(f(k)) = 2/3
f(k) = 2/3/(1-2/3) = 2.
k = 2/(1-2) = -2.

Answer A.

What is the general formula for finding f(k) through f(f(k))? I don't understand that process.

Hi erikvm,

In Function questions such as this, you're given the formula that you need to work with. As such, you have to think more about how the math "works" within the given formula.

Let's start with the function that we're given in this question:

f(X) = X/(X+1)

Next, let's run through a few quick examples of how this function "works":

IF....X = 1
f(1) = 1/(1+1) = 1/2

IF....X = 2
f(2) = 2/(2+1) = 2/3

IF....X = 3
f(3) = 3/(3+1) = 3/4

These examples are pretty simple, but they help to establish the math that you will eventually have to do to answer this question.

Now, let's make it a bit more difficult....

What if we take the f(f(2))? This means that we have to use the function TWICE - the first time with the number 2, the second time with the RESULT of the first calculation...

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

Since 2/3 is the result of the first calculation, we have to plug THAT value into the function to complete the second calculation....

f(2/3) = (2/3)/(2/3 + 1) = (2/3)/(5/3) = 2/5

So, the f(f(2)) = 2/5.

You should notice that the answers to this prompt are all numbers, so one of them MUST lead to the result (2/3) when you plug it into f(f(X)) = 2/3. Are you comfortable plugging in those answers and doing the math as outlined in the above example?

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

Thank you for this explaination. I do understand the process of plugging in a value "twice", so to speak. However, now I don't have a value to plug in. I simply know what its gonna equate. How do I know that f(f(k)) can be written as f(k/k+1) ?

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

f(k)= k/(k+1)
f(f(k))= (k/(k+1)) / ( (k/(k+1)) +1 )
= (k/(k+1))/ ((2k+1)/(k+1))
= k / (2k+1)
so, k / (2k+1) =2/3
-> k= -2

f(k) = k/(k + 1)

f(k/(k + 1)) = k/(k + 1)/[k/(k + 1) + 1], thus:

k/(k + 1)/[k/(k + 1) + 1] = 2/3

We see that since the denominators on both sides of the equation are equal to one more than their respective numerators, so

k/(k + 1) = 2

k = 2(k + 1)

k = 2k + 2

-2 = k

Answer: A

f(f(k)) can be written as f(k/k+1)

Now, again applying the given condition on f(k/k+1)

So, f(k/k+1)= k/k+1/k/k+1 + 1 = 2/3

Or,k/2k+1 = 2/3

Or, k = -2

Correct Ans A

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