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Posted on June 29, 2012 by Anthony Ritz

Today’s post is for all the GMAT (and GRE and SAT and ACT) students out there. Working with a student this week reminded me just how confused many test-takers are about what it really means for something to be a function and how often it is that these test-takers wind up frozen and unable to even begin to tackle function questions.

Ultimately, a lot of people just don’t understand what functions actually are.

Here’s what a function isn’t: a function does not necessarily involve numbers; a function is not its equation; a function does not have to have an equation at all; a function is not a cryptic secret handshake understandable only by mathematicians and 99^th-percentile test takers.

No, a function is really just this: “an abstract entity that associates an input to a corresponding output according to some rule” (so says Wikipedia). More simply, a function is a black box. You know the type: It’s big and squarish and opaque, and when you put something into the box at one end, the box whirs and clicks and beeps and shakes mysteriously until – at last – something pops out the other side. If you’ve watched much science fiction at all, you’ve probably seen something that fits the bill.

The only further requirement is that if on one occasion you throw a particular input (or set of inputs) into the box and get a particular output, then every time you throw that same input into the box you should get the same result. So if on one occasion you throw your cat into the box and out pops Chuck Norris riding a unicorn (I know, I didn’t see that coming either!), well, then, that’s exactly what you should get every time you throw your cat in the box.

An equation is simply a description of what happens inside the box. The equation is not, strictly speaking, the function, but we frequently gloss over this distinction given the close connection between the two. (It’s like holding up a picture and saying “this is my friend”; technically, the picture is not your friend, just as the equation is not your function, but everyone generally knows what you mean.) So an equation like $f(x) = x^2$ is just telling us that whatever is put into the function comes out squared. Want $f(3)$ ? Great – $3^2$ . Want $f(497)$ ? Sure – $497^2$ . $f(y)$ ? No problem – $y^2$ . $f(cat)$ ? $cat^2$ (whatever that is). Whatever you put in, it replaces the variable $x$ everywhere it appears. Students often balk at composition (plugging one equation into another), but it’s no different. $f(x+2) = (x+2)^2$ . $f(g(x)) = (g(x))^2$ . Whatever you have, this function gives it back squared.

Of course, a function doesn’t have to apply to numbers and it doesn’t have to have an equation. It is possible to create a function question that just gives a chart of possible input and corresponding output values. Consider the following hypothetical function question:

Values of functions f and g are given in the chart below.

$x$                    $f(x)$                $g(x)$
———————————————————
&                     ^                      %
#                      &                     ^
^                      @                    #
@                    %                     @
%                     #                      #

Which of the following answer choices is equal to $f(f(g($ # $)))$ ?

A. &
B. #
C. ^
D. @
E. %

To solve this question, we need to work – as always – from the innermost parentheses outward. The symbol # has been run through the function $g$ , then through $f$ , then through $f$ again. That means that we must first calculate $g($ # $)$ . Looking to the chart, find the row labeled # under the column heading $x$ . Read across to the column heading $g($ x $)$ to find that $g($ # $) =$ ^. Next, this result is run through the function $f$ . By the same process, we find that $f($ ^ $) =$ @. Finally, $f$ is applied once more. The chart states that $f($ @ $) =$ %, so we can at last conclude that $f(f(g($ # $))) =$ %. The answer is E.

Even if a function does have an equation, it may be notated strangely in an attempt to confuse the test-taker. Tests like the GMAT regularly invent symbols for made-up functions, and it is not unusual to see things like

“Let $f$ # $g$ $=$ $\frac{2(f+3)}{g^2}$ .”

“The symbol @ is defined such that $p$ @ $q$ $=$ $|3q-4p|$ for all real numbers $p$ and $q$ .”

“ $n^*$ represents the number of unique factors of $n$ .”

When the question subsequently asks for the value of $3$ @ $(4$ @ $2)$ , the student may balk; “but they didn’t give me $3$ @ $(4$ @ $2)$ !” But of course they did; the equation $p$ @ $q$ $=$ $|3q-4p|$ tells you everything you need to know. For $4$ @ $2$ , we see $4$ in the first position, where $p$ was in the given equation. So we replace $p$ with $4$ everywhere in that equation. By the same token, we replace $q$ with $2$ wherever it occurs. In other words, $4$ @ $2$ $=$ $|(3*2)-(4*4)|$ $=$ $|6-16|$ $=$ $|-10|$ $=$ $10$ . Moving outward, the expression $3$ @ $(4$ @ $2)$ becomes $3$ @ $10$ $=$ $|(3*10)-(4*3)|$ $=$ $|30-12|$ $=$ $18$ . The answer is $18$ .

Mixing functions and finding the value of $3$ # $(4$ @ $2)$ seems like it would be harder, but it really doesn’t have to be – recall that $4$ @ $2$ is $10$ , which simplifies the expression to $3$ # $10$ . Since $f$ # $g = \frac{2(f+3)}{g^2}$ , plugging in our values produces $3$ # $10$ $=$ $\frac{2(3+3)}{10^2}$ $=$ $\frac{12}{100}$ $=$ $\frac{6}{50}$ $=$ $\frac{3}{25}$ . The answer is $\frac{3}{25},$ or $0.12$ .

Handling tricky function questions on these tests requires little more than a clear understanding of the basic nature of functions and the step-by-step application of a simple process. What it doesn’t take is any particular level of genius. With a bit of practice, you too can summon Chuck Norris, Unicorn Knight. Okay, probably not. But you can get correct answers on important standardized tests. And isn’t that just as good?

For a bit of extra practice, consider the following question:

Given that

$f$ # $g = \frac{2(f+3)}{g^2}$ ,

the symbol @ is defined such that $p$ @ $q = |3q-4p|$ for all real numbers $p$ and $q$ ,

and

$n$ * represents the number of unique factors of $n$ ,

…What is $(20$ * @ $6)$ * # $3$ *?

Post your answers in the comments!

Tags: ACT, Algebra, Functions, GMAT, GRE, Math, SAT

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