Function Fun

Okay, maybe functions aren’t fun.  But succeeding on the hard questions isn’t exactly a downer.  So let’s look into them a little more:

There are two ways to symbolize functions on the SAT.  One is the way you learned in school:

f(x) = 3x + 2

Another way is to use a made-up symbol.

Let x* = 3x + 2

Both equations means the same thing.  So f(5) = 15 + 2 = 17, and 5* = 17 too.

Some SAT functions are a little strange:

Let a ^ b = true if a is a multiple of b; otherwise a ^ b = false.  Here, you input two mathematical values, but the output isn’t a number or variable – it’s just “true” or “false.”  10 ^ 2 = true, since 10 is a multiple of 2, but 10 ^ 3 is false.

f(x) = 5 simply means that the output is 5, regardless of what x is.  It’s a pretty trivial function, but it’s good for confusing students.

Drawn in Autosketch + pasting in words from Excel

Image via Wikipedia

 

 

 

 

 

 

 

Q. If f(x,y) = 2x + 3y, what is f(2y,2x)?

 

This confuses a lot of students.  Huh – isn’t x supposed to be on the left?  What’s going on here is something I call “windows and variables.”  The definition of a function contains windows, or “supervariables.”  Here, the definition is f(x,y) = 2x + 3y.  The function is only defined once.  Every other time you see it will be an example, which can contain numbers or variablesf(2,5) is easy – it’s just 2(2) + 3(5).  In other words, you plug the number 2 into the x window, and the number 5 into the y window.  The example f(2y,2x) contains the variables x and y, which have nothing to do with the windows x and y.  All you need to do is plug the variable 2y into the x window, and the variable 2x into the y window.  So f(2y,2x) = 2(2y) + 3(2x) = 4y + 6x.

Another way to think of this is that f(x,y) = 2x + 3y means “start with 2 values, and end up with twice the left value plus 3 times the right value.”

What about graphing functions?  Here’s the trick: don’t try to understand what the graph means.  Not many people have any idea what a graph means, unless it’s explained slowly and it’s truly simple to begin with.  So just consider one point at a time.  So if you’re shown a graph of f(x) and asked which answer shows the graph of f(x+2), just find a point on the first graph.  Suppose it contains the point (1,3).  Then you know that f(1) = 3, even though you don’t know what the function is.  And since f(x+2) uses the same function, f(1) still equals 3, so the correct graph will include the point (-1,3) (since f(x+2) will be f(1) when x = -1).  If more than one graph in the answers includes that point, just move on to another point.

 

The following is typical of the most challenging function questions on the SAT:

Q. Let f(x) = 3x + 1

If 2f(a) = f(3a), what is the value of a?

This might look scary to you, but it’s actually not very hard!  The illusion of difficulty comes from your unfamiliarity with the symbol f( ), so you should get rid of it as soon as possible.

Recall that 2f(x) just means 2 [f(x)].  And here, f(a) means “start with a, and end up with 3 times a plus 1.  So now you can translate the equation into

2(3a + 1) = 3(3a) + 1

And that’s just a plain old algebraic equation.  If this is a multiple choice question, you can just try the answers (substitute for a) until you find the one that works.  Or just solve:

2(3a + 1) = 3(3a) + 1

6a + 2 = 9a + 1

2 = 3a + 1

1 = 3a

1/3 = a

Now let’s check:  substitute 1/3 for a

2(1 + 1) = 3(1) + 1

2 X 2 = 3 + 1

4 = 4 Cool.

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