Calculus > 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=x^{2}. 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 x^{2} might be 0>∞. But obviously, not all functions are that simple. There are three methods: guess and check, graphing, and a table of values.

## How to Find the Domain and Range of a Function: Steps

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/(x^{2} – 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. IF you don’t have a graphing calculator, try this online one.

Step 4: Make a table of values (How to make a table of values on the TI89). 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.

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