Plain-old (non-symbol) function questions

Not all function questions have weird symbols, some are just vanilla f(x) type things. You've probably been working with the f(x) notation in school for some time now, but let's review some of the things you'll see over and over again on the SAT:

Interpreting function notation

One thing you're definitely going to need to be able to do is interpret function notation. For some questions, it's enough to remember that saying f(x) = x³, for example, is basically the same as saying y = x³.

For other questions, you're going to need to take that a bit further and identify points on a graph using function notation. Here's a quick cheat for you (with colors!). When you have f(x) = y, that's the same as an ordered pair (x, y). For example, if you know that f(4) = 5, then you know that the graph of the function f contains the point (4, 5). Likewise, if you know that h(c) = p, you know that the graph of function h contains the point (c, p). And so on. Basically, whatever is inside the parentheses (that's called the argument, if you care) is your x-value, and whatever's across the equal sign is your y-value. This is important. If you don't understand yet, read it over and over until you do. Might help to write it down. Just sayin'.

To make sure you've got this, think about what the following things mean. Once you've thought about them, hold your mouse over them (don't click, just hover...follow directions) to see what they're about.

• f(0) = 3
• p(12) = 0
• s(3) = r(3)

Nested functions (functions in functions)

source

Have you ever seen those dolls where you open them up and there are smaller ones inside? If you haven't, I've put a picture of some really artsy-fartsy ones there on the right. They're called Russian nesting dolls. Welcome to the world. Can I take your coat?

Anyway, sometimes the SAT will put a function inside another function to try to bamboozle you. Don't let yourself get stymied here. All you need to do is follow the instructions, same as you do with all other function questions. Let's look at an example:

17. If f(x) = x² - 10 and g(x) = 2f(x) + 3, what is g(√2)?
    
    
    (A) 7
    (B) 5
    (C) -5
    (D) -7
    (E) -13

Let's just take this one step at a time. First, let's take the √2 we're given and put it in for every x we see in g(x).

g(√2) = 2f(√2) + 3

Not so bad, right? Now let's replace the f(√2) with an expression we can actually work with. Remember that f(x) = x² - 10, so we write:

g(√2) = 2[(√2)² - 10] + 3

See how this is working? Now just simplify:

g(√2) = 2[-8] + 3

g(√2) = -16 + 3

g(√2) = -13

So our answer is (E). Awesome, right?

Interpreting Graphs and Tables

Say the picture above shows the function g(x) over a given range. Note that, although we have no equation for g(x), we still know a lot about it. We know, for example, that g(4) = -2. We also know that the y-intercept, g(0), is about -1. Think about the following, again, mousing over them once you know what's up to see if you're right:

• If g(a) = 0, how many possible values are there for a in the given range?
• When, in the given range, is g(x) < -4? 

You could also be presented with function information in table form. Peep this:

x	2	3	4	5	6
f(x)	13	18	25	34	45

Just like in the graph above, we can use this table to find points. For example, f(5) = 34. See if you can do the following:

• If f(p) = 18, p =
• What is f(6-4)?
• What is f(6) - f(4)?
• Holy crap those are different?
• Can you figure out what the function f(x) is?

Graph Translation

Sometimes the SAT likes to test you on whether you can figure out where a graph will move based on some manipulation of its equation. Usually, though, they won't give you the equation. They'll draw some crappy squiggly like the one above, call it g(x), and then ask what will happen to g(x+1).

I'll give you the rules for this, but I highly recommend reminding yourself of them with your calculator if you should need them on your SAT. It's very easy to set a simple function (like f(x) = x², which you've seen a million times) as your starting point, and experiment with your graphing calculator to see how graphs will behave based on various modifications. In fact, why don't you play with this widget a bit right now to see what happens? 

Here are those rules:

• f(x)+1 ⇒ (graph moves UP one)
• f(x)-1 ⇒ (graph moves DOWN one)
• f(x+1) ⇒ (graph moves LEFT one)
• f(x-1) ⇒ (graph moves RIGHT one)

Sample Questions!

13. If f(x) = 2x - 1, then f(10) - f(5) =
    
    (A) 5
    (B) 9
    (C) 10
    (D) 11
    (E) 19

16. The figure above shows the graphs of f(x) and g(x). If f(b) = g(b), which of the following could be the value of b?
    
    
    (A) -3
    (B) 0
    (C) 1
    (D) 7
    (E) 10
    
    
    
    
    
    
    

    x	1	3	5	7	9
    g(x)	-2	15	6	-3	-10

    
    
20. If f(x) = |g(x)|, and f(7) = t, then f(t) =
    
    (A) -2
    (B) 15
    (C) 6
    (D) -3
    (E) -10

Answers:
13. (C)
17. (A)
20. (B)