Domain and Range
Domain and Range
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.
Domain of a Function
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.
Finding the Domain of a Function
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:
  1. Division by zero. Division by zero is mathematically impossible. A function is therefore undefined for all the values of x for which division by zero occurs. For example, f(x) = 1/x-2 is undefined at x = 2, since when x = 2, the function is equal to f(x) = 1/0.
  2. Negative numbers under square roots. The square root of a negative number does not exist, so if a function contains a square root, such as , the domain must be x > 0.
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:
What is the domain of f(x) = ?
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:
What is the domain of f(x) = ?
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:
  1. The denominator would equal zero if x = 7.
  2. The quantity under the square root, x – 4, must be greater than or equal to zero in order for the function to be defined. Therefore, .
The domain of the function is therefore the set of real numbers x such that x ≠ 7.
The Range of a Function
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.
  • Absolute value. The absolute value of a quantity is always positive. So, in a simple case, f(x) = |x|, you know that f(x) must always be positive, and so the range includes only zero and positive numbers: . Never assume that any function with an absolute value symbol has the same range, though. The range of g(x) = –|x| is zero and all of the negative numbers: .
  • Even Exponents. Any time you square a number (or raise it to any multiple of 2) the resulting quantity will be positive.
Finding the Range
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.
What is the range of ?
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:
What is the range of ?
Tackle this example step by step.
  1. The absolute value restricts the range to 0 ≤ f(x) ≤ ∞.
  2. Add 4 to each bound of the range. This action only affects the lower bound: 4 ≤ f(x) ≤ ∞.
  3. Taking the square root once again affects only the lower bound: 2 ≤ f(x) ≤ ∞.
  4. Finally, divide the bounds of the range in half to determine the range of the entire function: 1 ≤ f(x) ≤ ∞.
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.
The Range of a Function with a Prescribed Domain
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:
f(x) = 2x2 + 4 for –3 < x < 5. What is the range of f(x)?
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.
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