GMAT Tip: Dealing With Functions

GMAT Tip: Dealing With Functions
Three simple rules will remove a lot of the confusion surrounding algebraic functions (Photograph by Getty Images)
Photograph by Getty Images

This tip on improving your GMAT score was provided by Brian Galvin at Veritas Prep.

One of the most common recurring themes on the quantitative section of the GMAT is abstraction. The math itself isn’t college-level—it’s mostly arithmetic, algebra, and geometry—but the way in which it is presented can drive examinees crazy. And there’s good reason for that. Real-world business problems don’t generally come with turn-by-turn GPS navigation to success; those problems are abstract and challenging, requiring leaders to work to even understand the parameters before they can start to plan a solution.

But not all abstraction is created equal. In a way, function questions (remember the f(x) = … setup from high school?) are “GPS” problems—they give you turn-by-turn directions, at least if you’re ready for them. It’s quite possible to become highly-functioning on function problems without a great deal of study. Here’s how:

Know what functions mean:

When you see an equation such as f(x) = x2 – 16, recognize that what it’s really saying is “whatever goes in the parentheses goes wherever x appears on the other side of the equation.” So f(5) = 52 – 16 (you just put the 5 wherever x appeared in the “instructions” section of the function) and f(y2) = (y2)2 – 16. The single most important thing to know about functions is how they work. No matter how abstract or confusing they might appear, the rule is that you take whatever is in the parentheses and plug it in wherever that variable is on the other side of the equation.

When dealing with multiple functions, use parentheses rules and work from the inside out:

Suppose you see a problem that asks: If f(x) = x2, what is f(f(x))?

Here you need to follow the rule from above—whatever goes in the parentheses gets squared per the instructions. The only problem is that you have two sets of parentheses, so where do you start?  Functions are just algebra, and if it were a straight algebra problem you’d know: work the parentheses from the inside out. So first you need to take the innermost parentheses and apply the instructions. The innermost f(x) becomes x2, leaving you with f(x2). And then apply the primary rule of functions again—whatever is in the parentheses in this case gets squared per the instructions, so f(x2) = (x2)2 = x4.

When in doubt, plug in small numbers to see how the function acts:

At their core, function problems are mainly exercises in abstraction; the notation and the use of multiple letters tend to throw people off much more than the math itself. To get a feel for what the function is really telling you to do, try plugging in numbers, instead of letters, so you can make the function a little more concrete. A common GMAT function setup looks something like this:

For which of the following does f(a)−f(b)=f(a−b)for all values of a and b?

(A) f(x) = x2
(B) f(x) = x/2
(C) f(x) = x + 5
(D) f(x) = 2x – 1
(E) f(x) = |x|

This problem is abstract: It includes four letters (f, a, b, and x) and each letter appears multiple times. Look at the actual “math.” The functions in the answer choices aren’t all that involved. The key here is breaking through the clutter. So try plugging in small numbers for a and b to see how the functions will set up. If you call a = 6 and b = 4, then (a –b) = 2,  and you can go through the answer choices to see how they work:

f(6) – f(4) =             f(2) =

(A) f(x) = x2 36 – 16 = 20            4    (NO MATCH)
(B) f(x) = x/2        3 – 2 = 1            1    (MATCH)
(C) f(x) = x + 5        11 – 9 = 2            7     (NO MATCH)
(D) f(x) = 2x – 1        11 – 7 = 4            3    (NO MATCH)
(E) f(x) = |x|        6 – 4 = 2            2    (MATCH)

So you’ve already eliminated three answer choices because you have proof that f(a) – f(b) does not always match f(a – b) for those choices. (Just one non-match proves that it doesn’t always match.) Through the exercise you should have identified something about the two remaining choices—for answer E, the math you used is identical to the math you used to get from a and b to a – b. You haven’t really “tested” it, especially given that absolute values become interesting when negative numbers are involved. So try making a or b (or both) negative. If a = 6 and b = -4, then:

(E) f(x) = |x|        6 – 4 = 2            10    (NO MATCH)

Since you’ve now disproven four of the five answer choices, B must be (and is) correct.

More important than this particular problem is the lesson within it: Functions are largely difficult because they’re abstract. But if you know how to follow function instructions—whatever is in the parentheses gets plugged in for the variable, work from the innermost parentheses out, and plug in small numbers to make the abstract concrete—you can undercut the primary difficulty of functions. When functions are no longer abstract, they are far less difficult.

For more practice, visit the Veritas Prep GMAT question bank where you can work through realistic GMAT questions and review detailed solutions.

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