If the operation @ is defined for all integers a and b by a@b = a + b – ab, which of the following statements must be true for all integers a, b and c?

I. a@b = b@a

II. a@0 = a

III. (a@b)@c = a@(b@c)

(A) I only

(B) II only

(C) I and II only

(D) I and III only

(E) I, II and III

Here is another challenging function question from the GMAT Prep Tests 1 and 2. You can solve this one algebraically or by picking numbers. my gut says: pick numbers (as we’ve done on a bunch of these GMAT prep function questions).

## Let’s pick some numbers

Let’s make:

A = 3

B = 5

C = 8

I’m just picking random numbers trying to avoid things that might create exceptions (1, 0).

## Take your time plugging values into the statements

I’d start with statement ii as it’s the easiest.

**II. a@0 = a**

a@b = a + b – ab

3 + 0 – a(0) = 3.

That checks out and you can see that it doesn’t matter what number you pick for a.

So let’s cross off any choices that don’t include ii. A and D are out. Remember to use easy statements first so you can narrow things down.

Next easiest is probably the first roman numeral.

**I. a@b = b@a**

You might be able to see without plugging the numbers in that the order doesn’t matter. Why? Because with addition and multiplication the order doesn’t matter and that’s all we have here. Let’s plug in the numbers to check.

a = 3 b = 5

(3 + 5) – (3)(5) vs (5 + 3) – (5)(3)

-7 vs -7

They are equal so I is a keeper.

That eliminates B. So only C and E left.

Now we have to test statement III.

**III. (a@b)@c = a@(b@c)**

a = 3 b = 5 c = 8

We can reuse some work. We know that a@b = -7.

(a@b)@c

(-7 + 8) – (-7)(8) = 1 – (-56) = 57

Now let’s do the second part.

a@(b@c)

Let’s break it up a little.

b@c = (8 + 5) – (8)(5) = 13 – 40 = -27

a@-27

(3 + -27) – (3)(-27)

-24 + 81 = 57

Bingo. Also works. You might be thinking: but, is it possible that there will be some combination that doesn’t work? Maybe, but in this scenario that is extremely unlikely.

**Correct Answer:** E

## Additional GMAT Function Practice Questions

These two questions are very similar to the above and on both you can practice picking numbers and plugging back in. They are a little different cosmetically in that they have the f(x) notation vs the symbol (@) but same deal.

For which of the following functions is f(a+b) = f(b) + f(a) for all positive numbers a and b?

For which of the following functions f is f(x) = f(1-x) for all x?

This is a very challenging function question that adds in a number properties/divisibility component