A relation is a set of ordered pairs. Relations can be represented in a variety of ways, including lists, diagrams, equations, and graphs. The domain of a relation is the set of the first coordinates of the ordered pairs in the relation. The first coordinate is often represented by the variable . The range of a relation is the set of the second coordinates of the ordered pairs in the relation. The second coordinate is often represented by the variable .
A type of relation that is particularly important is called a function. A function is a relation in which each element of the domain corresponds to exactly one element of the range.
The domain of a function is the set of the first coordinates of the ordered pairs. A value in the domain of the function is called the input. The independent variable represents the input values of a function.
The range of a function is the set of the second coordinates of the ordered pairs. A value in the range of the function is called the output. The dependent variable represents the output values of a function.
In a function, each input has a unique output. If is a function of , then every -value corresponds to only one -value.
|Relation That Is Not a Function||Relation That Is a Function|
|The mapping diagram represents
the set of ordered pairs . The relation is not a function because the value of 1 in the domain corresponds to both 5 and 7 in the range.
|The mapping diagram represents the set of ordered pairs . Each value in the domain corresponds to only one value in the range, so the relation is a function. Note that a value in the range can correspond to more than one value in the domain.|
A function can be named by a letter, such as . When the function is written as an equation, the dependent variable is sometimes written as rather than . Read as " of ." It represents the value of the function at . While functions are frequently called , they may be named with any letter to distinguish among several functions or describe the output.For example, three functions may be named , , and .
Domain and Range
When a function is represented by an equation, analyze the equation to determine all possible input and output values.
- For the domain, look for values that would make the function undefined. This includes values that would make the denominator of a fraction equal to zero or values that would make an expression under a square root have a negative value.
- For the range, look for any values that could not be the output of the function. For example, if an expression is squared or a square root, the value cannot be negative. A fraction with a nonzero numerator cannot be equal to zero.
Write the domain and range. Both the domain and range are all real numbers.Using inequality notation, the domain is written as:
1. Write the equation.
2. Replace each with the desired value or expression.
3. Simplify the resulting expression.
The input of a function can also be an algebraic expression. In this case, the output will also be an algebraic expression, not just a number.
Operations with Functions
Operations with Functions