Functions are relationships between two sets of numbers; each number you put into the formula gives you only one possible answer. An algebraic expression in one variable can be used to define a function of that variable. A function is denoted by a letter such as f or g along with the variable in the expression.

For example, the expression x³ - 5x² + 2 defines a function f that can be denoted by f(x) = x³ - 5x² + 2.

Once a function f(x) is defined, it can be thought of the variable x as an input and f(x) as the corresponding output. In any function, there can be no more than one output for any given input. However, more than one input can give the same output.

Function: A rule that turns each member of one set of numbers into a member of another set.

Independent Variable (input): The number you want to find the function of; the x in f(x).

Dependent Variable (output): The result of substituting the independent value into the function, f(x). (This is like y variable.)

Domain: The set of all possible values of the independent variable.

Range: The set of all possible values of the dependent variable.

Domain of Function

The domain of a function is the set of all numbers that can possibly be an input of a function, the x in f(x). You can think of the domain as the set of all possible independent variables values you can put into a function.

Unless a problem specifies otherwise, the domain of a function includes all real numbers, which means that the only numbers that aren’t included in the domain are numbers that aren’t real. 

• A real number can’t be a fraction with a denominator of 0, because then the number would be undefined.

• A real number can’t be an even-numbered root of a negative number. Even-numbered roots of negatives aren’t real numbers because any number that’s squared or has an even-numbered power can’t result in a negative number.

Range of Function

The range of a function is the set of all numbers that can possibly be an output of a function, the value for f(x). You can think of the range as the set of all possible dependent variables values that can come out of any particular function.

Just as the domain of a function is limited by certain laws of mathematics, so, too, is the range.

• An absolute value of a real number can’t be a negative number.

• An even exponent or power can’t produce a negative number.