Definition-of-a-Relation一个的关系的定义-课件

1、Definition of a RelationA relation is any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation, and the set of all second components is called the range of the relation. Definition of a RelationA relaExample:Analyzing U.S. Mobile-Phone Bills

2、 as a RelationSolutionThe 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 is56.21, 51.00, 47.70, 42.78, 39.43. Find the domain and range of the relation(1994, 56.21), (2019, 51.00), (2019, 47.70),

3、(2019, 42.78), (2019, 39.43)Example:Analyzing U.S. Mobile-Definition of a FunctionA 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 funct

4、ion 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. Definition of a FunctionA funcExample:Determining Whether a Relation is a FunctionSolutionWe begin by making a figure for each relation that shows set X, the domain,

5、and set Y, the range, shown below. 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)1234689DomainRange(a)Figure (a) shows that every element in the domain corresponds to exactly one element in the range. No two ordered pairs in the given relati

6、on have the same first component and different second components. Thus, the relation is a function.6891234DomainRange(b)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. Example:Deter

7、mining Whether aWhen is a relation a 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.Determine whether each relation is a function.S = (1,2), (3,4),(5,6),(7,8)Each first componen

8、t is unique, therefore S is a functionWhen is a relation a function?Function NotationWhen 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

9、 of the functions inputs and the range as the set of the functions 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 variabl

10、e, 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 asy = f (x) = 4 - x2. Function NotationWhen an equatExample:Evaluating a FunctionSolutionWe substitute 2, x + 3, and -x for x in the definition o

11、f f. When replacing x with a variable or an algebraic expression, you might find it helpful to think of the functions equation asf (x) = x2 + 3x + 5. If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3) c. f (-x) a. We find f (2) by substituting 2 for x in the equation.f (2) = 22 + 3 2 + 5 = 4 +

12、6 + 5 = 15Thus, f (2) = 15.moremoreExample:Evaluating a FunctionExample:Evaluating a FunctionSolutionb. We find f (x + 3) by substituting x + 3 for x in the equation.f (x + 3) = (x + 3)2 + 3(x + 3) + 5If f (x) = x2 + 3x + 5, evaluate: a. f (2) b. f (x + 3) c. f (-x) Equivalently,f (x + 3) = (x + 3)2

13、 + 3(x + 3) + 5= x2 + 6x + 9 + 3x + 9 + 5= x2 + 9x + 23.Square x + 3 and distribute 3 throughout the parentheses.moremoreExample:Evaluating a FunctionExample:Evaluating a FunctionSolutionc. We find f (-x) by substituting -x for x in the equation.f (-x) = (-x)2 + 3(-x) + 5If f (x) = x2 + 3x + 5, eval

14、uate: a. f (2) b. f (x + 3) c. f (-x) Equivalently,f (-x) = (-x)2 + 3(-x) + 5 = x2 3x + 5.Example:Evaluating a FunctionFinding a Functions DomainIf a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f (x) is a real number. Excl

15、ude from a functions domain real numbers that causedivision by zero and real numbers that result in an even root of a negative number. Finding a Functions DomainIf Example:Finding the Domain of a FunctionSolutionNormally it is safe to say the domain of a function is all real numbers. However, there

16、are 2 conditions which must be considered: 1)division by zero and 2)even roots of negative numbers. Consider the following functions and find the domain of each function: a. The function f (x) = x2 7x contains neither division nor an even root. The domain of f is the set of all real numbers.moremore

17、b. The function contains division. Because division by 0 is undefined, we must exclude from the domain values of x that cause x2 9 to be 0. Thus, x cannot equal 3 or 3. The domain of function g is x | x = -3, x = 3./Example:Finding the Domain ofExample:Finding the Domain of a FunctionSolutionContinu

18、ing c. The function contains an even root. Because only nonnegative numbers have real square roots, the quantity under the radical sign, 3x + 12, must be greater than or equal to 0.3x + 12 03x -12x -4The domain of h is x | x -4 or the interval -4, oo).Example:Finding the Domain ofProblemsEvaluate each function for the given values.F(x) = 3x + 7 F(4)b. F(x+1)c. F(-x)F(x) =f(