Difficult SAT questions on functions test to see if you are the master of your domain and range. Here are the keys to the kingdom.

The domain of a function is the set of inputs (x values) for which the function is defined. Consider the functions f(x) = x² and g(x) = ¹/_x. In f(x), any value of x can produce a valid result, since any number can be squared. In g(x), though, not every value of x can generate an output: When x = 0, g(x) is undefined. While the domain of f(x) is all values of x, the domain of g(x) is x < 0 and x > 0. The domain of the function h(x) = is even more restricted. Since a negative number has no square root, h(x) has a domain of x > 0.

To find the domain of a given function, first look for any restrictions on the domain. There are two main restrictions for function domain questions to look out for on the SAT:

There are easy-to-spot warning signs that indicate you should look out for either division by zero or negative numbers under square roots. The division by zero warning sign is a variable in the denominator of a fraction. The negative number under square roots warning sign is a variable under a square root (or a variable raised to the 1/2 power). Once you’ve located the likely problem spots, you can usually find the values to eliminate from the domain pretty easily.

You must be itching for an example. Allow us to scratch that itch:

f(x) has variables in its denominator: red flag for the possibility of division by zero. You may need to restrict the function’s domain to ensure that division by zero doesn’t occur. To find the values of x that cause the denominator to equal zero, set up an equation equal to zero: x² + 5x + 6 = 0. A quadratic equation. Ahoy! Factor it: (x + 2)(x + 3) = 0. So, for x = {–2, –3}, the denominator is zero and f(x) is undefined. The domain of f(x) is the set of all real numbers x such that x ≠ –2, –3. This can also be written in the form {x: x ≠ –2, –3}.

Here’s another example:

This function has both warning signs: a variable under a square root and a variable in the denominator. It’s best to examine each situation separately:

The domain of the function is therefore the set of real numbers x such that x ≠ 7.

A function’s range is the set of all values of f(x) that can be generated by the function. The easiest way to think about range is to visualize it on a graph. The domain, which is all the valid values of x in the function, is the x-axis, while the range, all the values of f(x), is the y-axis. Take a look at the following two graphs:

What values of the y-axis are reached on each graph? In the graph on the left, you can see that every possible value of y, from negative infinity to positive infinity, is included in the range. The range could be written as . Contrast this with the graph on the right, where the range is quite limited: Only the values between –1 and 1 are part of the range. So the range is .

There are two main warning signs of functions with limited ranges: absolute value and even exponents.

Calculating the range of a complex function is similar to finding the domain. First, look for absolute values, even exponents, or other reasons that the range would be restricted. Then adjust that range step by step as you run down the same checklist you use to find the domain.

In this case, the absolute value around |x – 3| screams out that the range of f(x) excludes all negative numbers: . |x – 3| is then divided by 2, so you have to divide the range by 2. But this division doesn’t actually change the range, since both zero and infinity remain unchanged when halved.

Now for a more complicated example:

Tackle this example step by step.

Note that addition, subtraction, multiplication, division, and other mathematical operations don’t affect infinity. That’s why it’s particularly important to look for absolute values and even roots. Once you can find a bound on a range that isn’t infinity, you know that the operations on the function will affect that range.

Another way that the range of a function could be restricted is if the domain is itself restricted. If the SAT is feeling particularly nasty, it’ll nail you with this kind of complicated domain and range question:

The first thing you have to realize is that there’s no reason to assume that the range of the function will be at its high and low points at exactly the bounds of the restricted domain. If you assume that the range of has its high point at 5 and its low point at –3, well, that’s exactly what the SAT wants you to assume.

Here’s where having a graphing calculator is immensely helpful on the new SAT. If you graph , you’ll see

You can see from this graph that the low point of the range comes when x = 0 and the high point comes when x = 5. Plug 0 and 5 into the function to get the low and high bounds of the function for the range –3 < x < 5. f(0) = 4. f(5) = 54. So the range is 0 < f(x) < 54.