Consider these two practice GMAT Quantitative problems:

1) Given f(x) = 3x – 5, for what value of x does 2*[f(x)] – 1 = f(3x – 6)

- (A) 0

(B) 4

(C) 6

(D) 7

(E) 13

- (A) –2

(B) 5/3

(C) 1

(D) 2

(E) 8

If you find these questions completely incomprehensible, then you have found the right blog post.

## Function notation

The GMAT Quantitative section will ask an occasional question about function notation. Here is a basic catechism about functions and what you need to know about them for the GMAT.

## What is a function?

A function is a rule, a “machine”, that takes an input and gives an output. When we are told the equation of a function, that equation makes explicit the rule this particular function is following. For example, for the function f(x) = 3x – 5, the rule is: whatever input x you give —- and that input could be any real number —- this function will multiply this input by 3 and then subtract five from the product: that difference is the output. If I put in an input of x = 2, then I get an output of 3(2) – 5 = 1, and the way we compactly write that fact with function notation is: f(2) = 1. In other words, an input of 2 gives an output of 1.

Notice — this is a very subtle issue. The x that appears in the equation of a function is a different sort of variable than the ordinary x of solve-for-x algebra. This x is what one might call a “formula variable”, like the a, b, and c in the quadratic formula. In other words, the x of function notation is not an x that is equal to only a single value; rather, it can be set equal to any value, any real number on the number line, when we want to plug that number into the function.

## Typical misunderstandings of function notation

When we write f(x), many people new to function notation will misinterpret this as multiplication — as if there’s a thing “f” times the variable “x”. That is 100% incorrect. A function is a process through which the input goes. Cooking is a process through which food goes. Puberty is a process through which people go. A function is a process acting on the number, and the nature of that action is outside the categories of simple arithmetic actions (add, subtract, multiply, divide).

Relatedly, the parentheses of function notation are mathematically inviolable. Nothing may pass through these parentheses. Again, this can be anti-intuitive, because when parentheses are used in ordinary notation, you can distribute through parenthesis, factor out, etc. Because a function is a different category of mathematical object, its parentheses are of a different nature. Thus

If you can simply avoid these mistakes and respect at all times the inviolability of the function’s parentheses, you will already be in better shape than a sizable portion of GMAT test takers.

## How a mathematician thinks about a function

In the above section, I discussed ways that folks new to functions might misinterpret function notation. Now, I am going to discuss how functions are seen by people who really understand them. Suppose we have the function f(x) = 3x – 5. Here’s what a mathematician looking at this function sees:

Where folks new to function just see the letter x, mathematicians see a “box”, an empty slot, a space that is, in some ways, analogous to an artist’s blank canvas. Anything that get plugged into the box on the left needs to get plugged into the box on the right. We can plug in numbers — any of the continuous infinity of real numbers on the real number line. We can also plug in algebraic expressions: If I put (2x + 7) into the box on the left, I need to put that exact identical expression into the box on the right. I can even put whole functions — the same function or an entirely different function — into the boxes. In fact, the list of mathematical objects that can be plugged into a function extends into far more sophisticated mathematical objects (matrices, differential operators, etc.) that are well beyond the realm of GMAT Quant. The GMAT, though, will expect you to know what to do if they give you, say, the function f(x) = 3x – 5, and then ask you, say, to plug in the expression 2x + 7:

f(2x + 7) = 3*(2x + 7) – 5 = 6x + 21 – 5 = 6x + 16

## Summary

If this is your first time encountering, or first time understanding, function notation, it is a worthwhile topic to practice, so that you are comfortable with it by test day. If you feel you have learned something from this, go back and try those two practice problems again before reading the solutions below. Also, here’s a practice question from our product:

3)

## Practice problem explanations

1) We have the function f(x) = 3x – 5, and we want to some sophisticated algebra with it. Let’s look at the two sides of the prompt equation separately. The left side says: 2*[f(x)] – 1 —- this is saying: take f(x), which is equal to its equation, and multiply that by 2 and then subtract 1.

2*[f(x)] – 1 = 2*(3x – 5) – 1 = 6x – 10 – 1 = 6x – 11

The right side says f(3x – 6) — this means, take the algebraic expression (3x – 6) and plug it into the function, as discussed above in the section “How a mathematician things about a function.” This algebraic expression, (3x – 6), must take the place of x on both sides of the function equation.

f(3x – 6)= 3*[3x – 6] – 5 = 9x – 18 – 5 = 9x – 23

Now, set those two equal and solve for x:

9x – 23 = 6x – 11

9x = 6x – 11 + 23

9x = 6x + 12

9x – 6x = 12

3x = 12

x = 4

Answer = **B**

2) There are several ways to approach this problem. One quick way is to notice that if x = 2, f(2) = 2/3. That’s not the answer, but it gives us a shortcut. If f(k) = 2, then we see that f(f(k)) = f(2) = 2/3. So, really, finding the value of k that satisfies the prompt equation really simplifies to solving the equation f(k) = 2.

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

Multiply both sides by the denominator.

k = 2*(k + 1)

k = 2k + 2

k – 2k = 2

–k = 2

Multiply both sides by –1.

k = –2

answer = **A**

Mike,

I don’t know if you still monitor these blogs, but I’ve had a really hard time wrapping my mind around these functions questions in my studies. For the second equation, I understand what you’ve done to get to f(k)=2 since 2 must be plugged in to f(x) to get 2/3. The part I’m missing is why setting k/k+1=2 helps us find the value of k. Rather, when you plug the function of k, or f(k) into f(x) you get 2/3, so what makes simply plugging the variable k, instead of f(k) into f(x) okay? I hope that makes sense.

Chris

Hi Chris 🙂

Happy to help! In this problem, we’re given the function

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

And we’re told that

f(f(k)) = 2 / 3

As you mentioned, based on this, we can determine that f(k) = 2.

At this point, it’s key to remember what f(k) is. We know the numerical value of f(k) = 2 from our analysis above. We can also express f(k) as a function, replacing x with k in the provided function f(x):

f(

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

k) =k/ (k+1)Therefore,

f(k) =

2 = k / (k+1)When we solve for k as outlined in the explanation, we find that k = -2. We can check this answer by plugging in k = -2 into the function f(k) and varying that f(-2) = 2:

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

As you can see, when k = -2, f(k) = 2, which matches what we found before 🙂

I hope this clears up your doubts! If not, please let us know 🙂

Hi Mike,

Sorry, I could have saved both of our time by being precise.

Good thing is that in this process of re-framing my question, the solution is clear to me now. 🙂 I tried to solve it the conventional way first .i.e substituting the terms in the ‘box’.

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

f(f(k)) = {k / (k+1)} / {(k / k+1) +1} = 2/3

Upon simplification, we get : k / (2k +1) = 2/3. => 3k = 4k +2 => k = -2.

As solution provided in the blog explains the quicker/efficient way to approach the problem, I probably got boggled initially by the inception (function inside function) sort of concept. 🙂 Now the solution is clear to me. Thanks.

Kumar.

Dear Kumar,

I’m glad you understood that the solution is more efficient. The way you approached the problem was mathematically correct but highly unstrategic. It’s not enough to be mathematically correct on the GMAT: you always have to be thinking in terms of strategy and efficiency. The GMAT routinely punishes students for pursuing a correct approach in a very conventional and unimaginative way. Does this make sense?

Mike 🙂

Yes.Thanks.:)

Dear Kumar,

You are quite welcome, my friend. Best of luck to you!

Mike 🙂

Dear Mike,

Can you explain why in the first problem, you plug in 3x – 5 into 2*[f(x)] – 13x – 6 (which i can comprehend) but then take f(3x – 6) and plug it into 3x – 5 ?

I thought the right part of the function equation will be 3 * (3x-5)-6 and not 3*[3x – 6] – 5.

I hope this question makes sense 😀

I think I have figured it out…

you only plug in 3x-5 in f(x). But here we have f(3x-6) . So the box is so to speak not empty.,,,

Dear Luke,

I’m happy to respond. 🙂 This is a very difficult thing conceptually, and about at the limit of anything the GMAT would expect you to know. You see, we have to keep straight which expression is INSIDE and which is OUTSIDE. The INSIDE expression is the one we plug in, the one that substitutes for the “empty box” of x. The OUTSIDE expression is the expression of the function itself: this is the one that provides, as it were, the outer framework for the calculation.

Also, notice that wording is very important here. In the first problem we are given f(x) = 3x – 5: that’s the OUTSIDE expression of the function itself. We then have to compute f(3x – 6). This means 3x – 6 is the INSIDE expression. The INSIDE expression gets plugged into the OUTSIDE expression, and the OUTSIDE expression provides the framework. We know

f(x) = 3x – 5

which means

f([__]) = 3*[__] – 5

That’s the outer framework, provided by the OUTSIDE function. Now, the INSIDE expression gets plugged into those blanks:

f(3x – 6) = 3*(3x – 6) – 5

The INSIDE expression fills the empty box.

Does all this make sense?

Mike 🙂

I went through the blog and found it mostly similar to the video lesson. My is with question number #2. Before you got (what I’m calling fancy) with the shortcut of plugging in 2, I’m having trouble doing it the old fashion way. Can you please explain how to do it with out and shortcuts and than I will try to do it with the shortcut.

I saw Kumar’s post and I could not understand how he got:

if; f(k) = k / (k+1).

f(f(k)) = {k / (k+1)} / {(k / k+1) +1} = 2/3

Upon simplification, we get : k / (2k +1) = 2/3. => 3k = 4k +2 => k = -2.

In this line {k / (k+1)} / {(k / k+1) +1} = 2/3 why divide {k / (k+1)} by {(k / k+1) +1}? I would have multiplied. And why is he adding +1 ?

Furthermore, you wrote in the blog post ” finding the value of k that satisfies the prompt equation really simplifies to solving the equation f(k) = 2.”

Im not understanding that. If the question is solve for k and we know that f of f of k equals 2/3, even if f of k equals 2 what happened to 2/3 at the end of the question?

I hope this makes sense.

Hi Mike,

Solution to Question 2 seems to be a bit confusing. Can you please elaborate more on it.

TIA.

Kumar.

Kumar,

First of all, I am going to suggest that you read this blog very carefully:

http://clemmonsdogpark.info/gmat/2014/asking-excellent-questions/

What you asked here is not a very high quality question. If you learn to ask better questions, you will get much more from your studies and you will progress more quickly. I challenge you to raise the quality of your question, and then I will answer it.

Mike 🙂

It’s understandable to be a little bit confused by Kumar’s longhand approach. As both Kumar and Mike discussed, that longhand approach is a bit complicated– needlessly so, really!

Basically, what Kumar is doing is taking x = x/(x+1) and turning it into k = k/(k+1).

Now, if k = k/(k+1), then the instances of k inside of the notation k/(k+1) should be changed into k/(k+1). If this still sounds confusing, think of it this way:

k= k/(k+1)

Therefore, you should think of the notation k/(k+1) as:

[k = k/(k+1)]/{[k= k/(k+1)]+1}

or to remove the equal sings and just plug int he value of k:

[k/(k+1)]/{[k/(k+1)]+1}

Hopefully that makes sense. But like Kumar, Mike, and I said, this is an unnecessarily complicated approach. So you may not want to ruminate on it too much. 🙂 (Still, let me know if you have more questions.)

Finally, you asked:

“If the question is solve for k and we know that f of f of k equals 2/3, even if f of k equals 2 what happened to 2/3 at the end of the question?

I hope this makes sense.”

If I understand correctly, you’re wondering how k can equal 2, but f of k does NOT equal 2? If that’s what you’re asking, it’s because k is a component within f of k. f of k and k would not equal the same thing.

Instead, f(k) would equal k/(k+1). If f(k) also equals 2/3, then the k in k/(k+1) must be 2. Because if you plug in 2 for k, you get 2/(2+1), or 2/3.