6.1 SAT Math - Functions

What follows in this section is most of the weird-looking questions or the ones that will confuse you if you don't have a little background knowledge.  Some, like functions, will show up a lot and are important to know, but quite a few are also, thankfully, pretty rare. 

Functions

Functions aren't rare at all, and they're not that weird either, but the way they're tested on the SAT can be.  First of all, what does a function look like?  A function can be graphed as a line, and we'll cover the graphing of functions in the Geometry sections, but for algebra purposes, a function is simply an equation.  3x = 2y + 4 is definitely a function; so is 4 = y + √x.  The difference between functions and regular equations here is that functions will be written as f(x), g(x), or the like.  That f(x) replaces the y value once the equation is solved for y.  So using the two examples above, in the first, solved for y, that equation should read  and the second would read y = 4 -√x.  But instead of writing that, we write

-Hint: Always remember that f(x) is the same thing as y!  This will help a lot when we get to graphing.

So now that we have what it looks like, what is it?  A function is a series of steps for you to perform.  That's it.  Think of it like a button on your calculator.  If you type the "squared" button on your calculator, it does an operation – it multiplies whatever number you gave it by itself.  Well, the "function" button would do the operation given you in one of these problems – so you'd type in a number and then hit "f(x)" and it would spit out an answer.  That's what you're doing with function problems.  They'll give you the definition of the function, that is, what steps you need to perform, and then they'll say "when x = 3" or some such, and you'll use 3 for x and solve. 
Why is this any different than the basic substitution you were doing earlier?  Because of the way they write it.  They don't actually say "when x = 3" usually.  Usually they'll have something like this:

11. If f(x) = 4^x + x², what is f(4)?

The number in the parentheses after the f is the number you will use for x.  Put it into the equation wherever you see an x and solve:

f(4) = 4⁴ + 4²

f(4) = 272

-SAT Math Tip:  Put whatever is in the parentheses in for x, no matter what it is!  So f(3)?  Use 3. f(x + 2)?  Put in x + 2.  f(shoe)?  Put in the word "shoe."  Keep parentheses around whatever you sub in for x, however, as it does represent one number. 

Example Problem

Try this SAT math question:

What did you get?  If you remembered to keep your parentheses and simply sub in (x - 4) for x, you should have gotten (D).

Domain and Range

Let's talk about domain and range.  The domain of a function is what you put into the equation in order to get an answer.  Basically, it means the possible x values.  Range is what you get out of the equation, or, your y or f(x) values.    So if they're asking for the domain of the question, they're asking "what values of x are possible in this equation?"
Sometimes the test will specify a particular domain or range and ask you to find the other.  If that's the case, simply plug in the endpoints of the specified domain to find the range:

7. If 2 ≤ x ≤ 6, what is the range of f(x) = 3x + 2?

Simply put in the endpoints given for x, in this case 2 and 6, and see what you get:

So the answer would be 8 ≤f(x) ≤ 20.
-SAT Math Tip: Be careful of values that can affect your domain or range!  Remember your previous rules – absolute values and square roots are positive only, a zero can not appear in a denominator, etc.  If the question you're given might be testing these rules, throw in other tricky numbers included in your domain also.  Let's take a look at one example:

If you throw in the two endpoints, you will get 4- 5 = -1 and 4 -5 = -1, and you may be tempted to answer f(x) = -1, but if you remember that you're dealing with a range of answers, you'll remember to try some odd choices:
-SAT Math Hint: Always try 0, 1, negatives and fractions.  These will test the limits of your range.

So what happens if you try 0 for x?  You'll get -5 for an answer, increasing your range from simply x = -1 to -5 < f(x) < -1.  If you try 1 or fractions, your range will also expand, though in this case, -5 is the limit.
-SAT Math Hint: Eliminate wrong answer choices as you go so you don't waste time looking for expanded ranges that aren't even an option.  In this case, as soon as you found the first limit of -1, you should have crossed out (A) because it doesn't include -1 in its values.  Once you found a viable value of -5 for f(x), you can cross out C), D) and E), leaving you with only B) as an option.  You never needed to try 1, negatives or fractions in this case because you narrowed your answer to the only remaining one so quickly.

Starting SAT Scores of 600 and Above:

Sometimes you will not be given a specified domain for the function.  In almost every case on this test, when that occurs, you want to focus on finding the function's minimum value.

If you are not given a specified domain for a function, there is still a way to find the range.  In these questions, you will generally be given a parenthetical expression with an even exponent (like (x -3 )²⁾, a square root expression (such as ) or an absolute value expression (like |x - 3|).  What you'll need to do to find the range is change the equals sign to a greater than or equal sign (≥) and then replace the expression with zero (the whole expression, not just the variable) and simplify.  So if you were given f(x) = (x - 1)² + 3, you would write f(x) ≥ 0 + 3, and your answer would be f(x) ≥ 3 or all real values ≥ 3.

If you have a negative coefficient in front of any of those expressions, you'll need to anticipate the rule about multiplying or dividing by a negative in an inequality and switch the sign to ≤.  So for  would be solved by changing that equals sign to a less than or equal sign, then changing the expression to zero, then simplifying: f(x) ≤ -4 (0) -3, simplified to f(x) ≤ -3 or all real values ≤ -3.

Sometimes you'll find that a function has no solution or is undefined.  On this test, that simply means find the value or values of x that will make the denominator equal to zero.  So all you need to do is set the denominator equal to zero and solve:

This function is undefined at x = -(1/4).

The other way they'll test when a function is undefined is by the "vertical line test".  On a graph of a function, if you draw a vertical (up and down) line through any part of the graph, you should only hit one point.  If you hit two points, it's not a function.  We'll talk more about graphing functions when we cover graphing, but try this vertical line test question:

9. Which of the following represents a graph of a function of x?