How to Find the Domain of a Function
The domain of a function is the set of numbers that can go in to a given function. In other words, it is the set of x-values that you can put in to any given equation. The set of possible y-values is called the range. If you want to know how to find the domain of a function in a variety of situations, just follow these steps.
Method One of Six:
Learning the Basics Edit
Learn the definition of the domain. The domain is defined as the set of input values for which the function produces an output value. In other words, the domain is the full set of x-values that can be plugged into a function to produce a y-value.
Learn how to find the domain of a variety of functions. The type of function will determine the best method for finding a domain. Here are the basics that you need to know about each type of function, which will be explained in the next section:
- A polynomial function without radicals or variables in the denominator. For this type of function, the domain is all real numbers.
- A function with a fraction with a variable in the denominator. To find the domain of this type of function, set the bottom equal to zero and exclude the x value you find when you solve the equation.
- A function with a variable inside a radical sign. To find the domain of this type of function, just set the terms inside the radical sign to 0 and solve to find the values that would work for x.
- A function using the natural log (ln). Just set the terms in the parentheses to 0 and solve.
- A graph. Check out the graph to see which values work for x.
- A relation. This will be a list of x and y coordinates. Your domain will simply be a list of x coordinates.
Correctly state the domain. The proper notation for the domain is easy to learn, but it is important that you write it correctly to express the correct answer and get full points on assignments and tests. Here are a few things you need to know about writing the domain of a function:
- The format for expressing the domain is an open bracket/parenthesis, followed by the 2 endpoints of the domain separated by a comma, followed by a closed bracket/parenthesis.
- For example, [-1,5). This means that the domain goes from -1 to 5.
- Use brackets such as [ and ] to indicate that a number is included in the domain.
- So in the example, [-1,5), the domain includes -1.
- Use parentheses such as ( and ) to indicate that a number is not included in the domain.
- So in the example, [-1,5), 5 is not included in the domain. The domain stops arbitrarily short of 5, i.e. 4.999…
- Use “U” (meaning “union”) to connect parts of the domain that are separated by a gap.’
- For example, [-1,5) U (5,10]. This means that the domain goes from -1 to 10, inclusive, but that there is a gap in the domain at 5. This could be the result of, for example, a function with “x – 5” in the denominator.
- You can use as many “U” symbols as necessary if the domain has multiple gaps in it.
- Use infinity and negative infinity signs to express that the domain goes on infinitely in either direction.
- Always use ( ), not [ ], with infinity symbols.
Check out the x-values that are included in the graph. This may be easier said than done, but here are some tips:
- A line. If you see a line on the graph that extends to infinity, then all versions of x will be covered eventually, so the domain is equal to all real numbers.
- A normal parabola. If you see a parabola that is facing upwards or downwards, then yes, the domain will be all real numbers, because all numbers on the x axis will eventually be covered.
- A sideways parabola. Now, if you have a parabola with a vertex at (4,0) which extends infinitely to the right, then your domain is D = [4,∞)
State the domain. Just state the domain based on the type of graph you’re working with. If you’re uncertain and know the equation of the line, plug the x-coordinates back into the function to check.