Hi NaimaB,

This question can be solved algebraically or by TESTING THE ANSWERS. Before we get to the solution, here's some perspective on what all of this "means":

f(X) = X/(X+1) can be graphed (it's essentially just a "plug in" prompt). Plug in a value for X and you get a result.

For example:

f(0) = 0/(0+1) = 0/1 = 0
f(1) = 1/(1+1) = 1/2
f(2) = 2/(2+1) = 2/3
Etc.

This question comes with the added "quirk" that you have to use the function TWICE: f(f(K) = 2/3 means that you plug in a number, then plug in the RESULT of THAT calculation and you end up with 2/3.

I mentioned earlier that you can solve this problem with algebra or you can TEST THE ANSWERS. Since I've already done some math, I'll focus on that approach. From the above examples, you can see that....

f(2) = 2/3

2/3 is the result that we're looking for, but we need to use the function TWICE and here we've only used it once. So now we have to figure out how to get 2 as a RESULT:

X/(X+1) = 2

X = 2(X+1)
X = 2X + 2
-2 = X

So, f(f(-2) = 2/3

We can test this out by running through the calculations:

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

Now we plug in the "2":

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

This is a match for what we're looking for!

Final Answer: A

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

I did not understand the question. From where this k comes?

This is something that a lot of students get confused about - the structure of function questions can be very tricky.

If the GMAT says:
If f(x) = 2x + 2, and f(m) = 8, what is the value of m?

Here's how to break down what this means:

f(x) means "the function that we perform on any given number x."

f(m) or f(k) or any other variable means "this function applied to the specific unknown number m (or k or whatever)."

So when they give you f(x), it's the general rule applied to any x. When they follow it up with f(any other variable), that variable is referring to a SPECIFIC number.

So here, f(f(k)) means "the function of the function of the specific number k." First, we'd figure out how to set up f(f(x)). We plug the whole function f(x) in as x:


Then, we're told that that function of k is equal to 2/3. Plug in k and simplify:


When we cross-multiply from there, we're left with:

3k = 2(2k + 1)
3k = 4k + 2
-k = 2
k = -2

The answer is A.