How to Find the Domain and Range of a Function
The domain is the set of x- values that can be inputted into a function, while a range is the set of y- values that are outputted for the domain. Many functions have an infinite set for the domain: for example, you could input any number you like into the function y=x2. But what about the range? Clearly, there will never be a negative number outputted for this function (because a negative times a negative will always be positive). So you could take a guess that the range for x2 might be 0>∞. But obviously, not all functions are that simple. There are three methods: guess and check, graphing, and a table of values.
Step 1: See if you can determine what type of function you have (this isn’t always clear!). Certain functions have defined domains and range.
- Linear functions and polynomial functions have the domain and range of all real numbers
- Square (quadratic) functions and absolute value functions have a domain of all real numbers and a range of y≥0
- Square root functions have a domain of x≥0 and a range of y≥0
- Rational functions have a domain of x≠0 and a range of x≠0.
- Sine functions and cosine functions have a domain of all real numbers and a range of -1 ≤y≥ 1.
Step 2:Try to use the guess and check method to determine the domain first. You’ll need a set of strong algebra skills to perform this step. For example, if you have the function f(x)=1/(x2 – 9), you can exclude any values of x (the domain) that make the denominator equal to zero (because division by zero is not defined).
Step 3:Graph your function and see where your x-values and y-values lie. Most graphing calculators will help you see a function’s domain (or indicate which values might not be permissible. For example, if you graphed x^2, it would be clear that the domain cannot include negative numbers.
Step 4:Make a table of values. Include inputs of x from -10 to 10, then some larger numbers (like one million). Use a calculator to find values of y for values of x. If the calculator tells you the values or undefined, or that the values might be reaching a limit (a number that a function approaches, but never reaches), that should help you determine the range.
Tip: Become familiar with the shapes of basic functions like sin/cosine and polynomials. That way, you’ll be able to reasonably find the domain and range of a function just by looking at the equation.