The SAT has lot of moves that look way scarier than they actually are (I fall for these fake scary problems every time, by the way).

I can not be the only one who chokes every time I see a                                                             “F U N C T I O N  P R O B L E M”                                                                                          (right?)

The other day I wrote a blog post about about my “function distress.” I was certain that I’d studied the concept “to mastery,” and had therefore moved on to the other mountains that needed moving.


Turns out my mastery was of the fleeting variety.

Thankfully, the wonderful world of the internet intervened in the form of solutions from generous commenters.

Herewith, that gnarly problem again, from the May 2011 SAT — and a variety of commenter solutions:


1) This first explanation comes from PC Keller, the author of The New Math SAT Game Plan.  Incidentally, many many smart people have told me they finally learned how to do a function problem after reading his book. 

The SAT draws on standard archetypes repeatedly — and this is one of them:  the funny-shaped graph that does not correspond to any single function you know (usually becasue it is designed “piece-wise” but you don’t need to know that…)

When you see one of these, here’s what you want to pop into your head immediately:

1.  Functions have inputs that get assigned unique outputs

2.  Inputs are along the x-axis

3.  Outputs are along the y-axis

With that in mind, the choices can be translated:

  •  I  When the x value is b, the y value is zero
  • II  When the x value is a, you get a bigger y value than    when the x-value is c
  • III The two y-values you get when you use x = a and x = 0 together add up to zero.

Notice that once you understand how to read a function graph, deciding which of these is true requires no more math than counting boxes!


2) The next solution was left in the comments by “Postijen.”  I am assuming she is a teacher from her other comments.

It’s those letters (parentheses) that make it look like it’s insanely difficult!   And just to make it that tiniest bit harder, they put the y first up there in the graph and used ‘g’ instead of how you usually see f(x) = y   All those things combine to make you think this is some new and horrible problem.  It’s not.

g(x) = y  just means (for the purposes of understanding these problems on the SAT, I’m no mathematician!)  that when you put in a value for x you will get that y.  Or in English  When x is ___ then y will be ____.

So in the problem above:

Choice I says: when x is b, y is 0 (true or false?) –>  find b on the x-axis of the graph, go up to the line, uh, yup, where x is b, y is 0, True  (Cross off B)

Choice II says when ‘x’ is ‘a’, ‘y’ is bigger than ‘y’ when ‘x’ is ‘c.’  Check it out — at ‘a’, ‘y’ is 2 and at ‘c’ ‘y’ is 1.  2 > 1  Yupper!  (Cross out A and D)

Choice III translates to ‘y’ when ‘x’ is ‘a’  plus ‘y’ when ‘x’ is ‘0’  so 2 + -2 = 0 True again! (Left with E as answer)



3) And finally, for the dyscalculics of the world, a lovely non-mathy variety by “Agjacobson”:

If they give you a graph of a function, it’s a wonderful thing. You just have to look at the graph. It’s like a phone book with all the numbers right on a diagram. You can look up all the values. So when the problem refers to g(a), you find a, then look up where g(a) is. Go to a, then look up or down over a to see where the graph crosses x=a, and that’s g(a). It’s like looking up a’s phone number g(a). The phone book is also a kind of function where the input is a name and the output is a 10-digit code. All functions involve either computing a value, or looking up a value. Then, when you get the value, you can test the assertions of the problem.


llustrations by Jennifer Orkin Lewis