Definition of a Relation一个的关系的定义-19页精选文档

function is a correspondence between two sets X and Y that assigns to each element x of set X exactly one element y of set Y. For each element x in X, the corresponding element y in Y is called the value of the function at x. The set X is called the domain of the function, and the set of all function values, Y, is called the range of the function.
• T = {(1,2), (3,4),(6,5),(1,5)} Note that the first component in the first pair is the same as the first component in the second pair, therefore T is not a function.
8 9
Domain
1 2 3 4
Range
Figure (a) shows that every element in the domain corresponds to exactly one element in the range. No two ordered pairs in the given relation have the same first component and different second components. Thus, the relation is a function.
Solution We begin by making a figure for each relation that shows set X, the domain, and set Y, the range, shown below.
(a) 1 2 3 4
Domain
6 8 9
Range
(b) 6
Figure (b) shows that 6 corresponds to both 1 and 2. This relation is not a function; two ordered pairs have the same first component and different second components.
Function Notation
When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H. Any letter can be used to name a function. Suppose that f names a function. Think of the domain as the set of the function's inputs and the range as the set of the function's outputs. The input is represented by x and the output by f (x). The special notation f(x), read "f of x" or "f at x," represents the value of the function at the number x.
If a function is named f and x represents the independent variable, the notation f (x) corresponds to the y-value for a given x. Thus, f (x) = 4 - x2 and y = 4 - x2 define the same function. This function may be written as
Example:Analyzing U.S. Mobile-Phone Bills as a Relation
Find the domain and range of the relation {(1994, 56.21), (2019, 51.00), (2019, 47.70), (2019, 42.78), (2019, 39.43)}
When is a relation a function?
Determine whether each relation is a function. • S = {(1,2), (3,4),(5,6),(7,8)}
Each first component is unique, therefore S is a function
Solution The domain is the set of all first components. Thus, the domain is {1994,2019,2019,2019,2019}.
The range is the set of all second components. Thus, the range is {56.21, 51.00, 47.70, 42.78, 39.43}.
Example: Determining Whether a Relation is a Function
Determine whether each relation is a function. a. {(1, 6), (2, 6), (3, 8), (4, 9)} b. {(6,1),(6,2),(8,3),(9,4)}