You’re going to find about 2-6 function questions on your SAT, and if you haven’t been working with them recently in math class, they might throw you for a loop. Just to be clear, we’re talking about equations that look a bit like this:
And that format might bring to mind a couple of things. You might be getting ready to graph it out off the bat, which is great, but not necessarily what you’ll need to do on the SAT. Or if you’re not so familiar with the type of equation, you might make a lethal error and assume the g here is a variable. It’s not, by any means, but…
…these functions aren’t always a separate skill. They’re just wearing costumes.
Some good news: the majority of SAT function questions are actually designed to test the same skills as non-function equation questions. The biggest difference is that they come later on in the section, being categorized as harder, which is great; that’s a medium to high-difficulty question that’s solvable in low-difficulty time.
Let’s take a look at an example of a relatively simple function question.
If , and , what is the value of a?
g(x) is basically just another variable, here. Let’s replace it with a y. As an equation question, it would look like this:
If , at what value of x does ?
The algebra from here on will bring you to the same result for either question (because they’re the same question!).
That g is just shorthand
Let’s say you have a neurotic friend. Most of us have one or two… but let’s go even more neurotic than whoever you have in mind. More toward obsessive-compulsive. Let’s call her Gina. Gina brings grapes to school every day, but she has a really bizarre habit: she eats four, splits the rest into two equal piles, then throws away one of piles and gives away what’s left. The thing is that she brings a different number of grapes every day. What she gives away at the end, then, also varies. You might recognize this process from above… it’s just an example of that earlier function g(x).
x is how many grapes Gina comes to school with. Everything done to x (on the right side of the equation) just details her bizarre habit. What she has left at the end—what she gives away—is g(x) or y, depending on the equation.
What that g really means is to follow Gina’s process with the number x . She brings x grapes, and she then gs the grapes. In a way, it’s a verb, not a noun. It indicates the process, not the piece, without writing out each step of the way.
Functions inside functions?
So let’s say you’ve got a whole group of neurotic friends. Gina does her grape ritual and then hands off what’s left to Hailey. Hailey gets a bit weird, too, and goes through a whole other process. She won’t eat them unless she gets the same amount of grapes from somebody else, and she then squashes one of them under her foot. So Hailey’s equation would be h(x) = 2x – 1. What she has, then, is described by h(x). But first we need to think about how many Gina gave away.
So let’s put the two processes together into one: h(g(x)). Take it from the inside out, and you’ll have no problem. Gina brought 20 grapes? First find how many she’ll end up giving away with her function, g(x), then put it through the process that Hailey uses, h(g(x)), to find out how many end up going down her gullet (it would be 15). More friends with more weirdo habits? Just keep nesting the function, e.g. f(h(g(x)).
In the end, SAT functions like this will end up being no different than the simpler-seeming SAT equations.
Coming soon: more about functions
If this hasn’t quite resolved your function fears, don’t fret. There’s more to cover—matching functions to their graphs, function transformations, and SAT symbol questions—so keep your eyes on upcoming posts. But it’ll be the same message, in the end: they’re totally doable.
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About Lucas Fink
Lucas is the teacher behind Magoosh TOEFL. He’s been teaching TOEFL preparation and more general English since 2009, and the SAT since 2008. Between his time at Bard College and teaching abroad, he has studied Japanese, Czech, and Korean. None of them come in handy, nowadays.
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