Domain and Range

Section 1.1 is an introduction to functions. To understand the concepts of domain and range, you must first understand functions. A function is a relation that matches each item from one set with exactly one item from a different set.

For example:

is a function. For any input, x, there is exactly one output, f(x).

The domain of a function is the set of all values of the independent variable for which the function is defined. For every value that is in the domain, f is defined at x. For every value that is not in the domain, f is undefined at x.
is the domain of the function in the example above because for any value of x, the function is defined.

The range of a function is the set of all values assumed by the dependent variable.

is the range of the function in the example above because for any input, the output will be at least 4.
is nonnegative. Therefore, the lowest possible value of f(x) is when x = 0, and the output is 4. Furthermore, as the inputs approach positive infinity and negative infinity, the outputs approach      positive infinity.



When determining the domain of a function, there are a few things to look out for. The first is dividing by zero. Any time a function includes one or more fractions, the domain excludes any value that would make the denominator of one of the fractions equal to zero. An input that makes the denominator zero makes the function undefined at that input value.

For example:




The domain of this function excludes x = 2 and x = -2. Both of these values make the denominator zero and therefore make the function undefined.
 is the domain of this function.


Another thing to look out for when determining the domain of a function is an even root of a negative number because the answer is imaginary.

For example:



The domain of this function excludes all numbers that make the value under the radical negative.
 is the domain of this function.

However, odd roots can have a negative value under the radical. When a negative number is raised to an odd power, the result is a negative number.

For example:



Likewise,




Finally, when determining the domain of a function, it is important to keep in mind what the function represents.

For example, if P(x) = 5x, where P represents a company's profit and x represents the number of units sold, then the domain is

because the company cannot sell a negative number of units.