RSS Feed 
How to Understand the Domain of Functions
Is your instructor talking Greek when he talks about the domain of a function? This article will help explain domain in English.
- Step 1
First, let's understand real-valued functions. We can think of functions as sets of ordered pairs (or triples or n-tuples) or as special rules. For the sake of simplicity let's think of functions as rules. These rules pair one number to some input number or they pair one number to some list of numbers. Again for the sake of simplicity, let's limit our discussion to the case where the inputs are single numbers. Hence, a function is a rule that pairs a number with another input number. For example, the function y = 2x pairs a number with its double. If we put "in" 6 for "x," the function says y = 12. Hence, 12 is paired with 6. We write this pair as an (x,y) ordered pair: (6,12). Frequently, books let y = f(x) and then write f(x) = 2x. This gives us a notation for "putting in 6," which is f(6). The notation f(6) refers to the number that will be paired with 6 by the function. In this case, that number is 12; hence, f(6) is 12.
- Step 2
Second, let's understand domain. A domain is a set of numbers associated with a function. Specifically, the domain of a function is the set of input numbers to which the function can assign a real value. In our example above where we have f(x) = 2x, the inputs include any real number since any real number has a double. Hence, the domain is the set of all real numbers.
- Step 3
Third, let's think about a case where a rule might not be able to pair a number with a potential input. For example, let's consider a rule that pairs an input with its reciprocal, i.e., f(x) = 1/x. This rule would pair 4 with 1/4, 2/3 with 3/2, and -7 with -1/7. There is one number, however, that does not have a reciprocal. That number is zero. Since the reciprocal function cannot pair a number with zero, we call zero a restriction on the domain of the reciprocal function. Hence, the domain of f(x) = 1/x is all real numbers except zero.
- Step 4
Fourth, let's recognize the three most common rules that restrict the domain. 1. Any rule that involves division might have domain restrictions. The reciprocal function above is an example. 2. Any rule that involves taking a square-root (or any even-numbered root) might have domain restrictions since the radicand of a square-root must not be negative. 3. Any rule that involves logarithms might have domain restrictions since the argument of a logarithm must be positive.
- Step 5
Finally, let's find the domain of three specific functions with restrictions.
1. Let's look at y = x/(x^2 - 4). This function involves division, and division is not defined whenever the divisor (the denominator) is zero. Hence, to find domain restrictions, we solve the equation below.
x^2 - 4 = 0
(x + 2)(x - 2) = 0
x = -2 or 2
These are x-values that will render the denominator zero, but a denominator of zero is not allowed. Therefore, the domain of this function includes any number other than -2 or 2.
2. Let's consider f(x) = square-root(4-x). This function involves taking the square-root, requiring that the radicand be non-negative. Hence, to find the domain, we solve the equation and inequality below.
4-x = 0
-x = -4
x = 4
4-x > 0
-x > -4
x < 4
Therefore, the domain of f(x) must be equal to zero or less than zero.
3. Let's consider y = log(2x+5). This function involves a logarithm, but the argument of a logarithm must be positive. Hence, to find the domain, we set the argument greater than or equal to zero and solve as below.
2x+5 > 0
2x > -5
x > -5/2
The domain of y = log(2x+5) includes all real numbers greater than -2.5.
