函数Function

1、1Section 1.8Function2Definition 1 Let A and B be sets. A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a)=b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B,we write f : AB3Definiti

2、on 1Let A and B be sets. A function (mapping, map) f from A to B, denoted f :A B, is a subset of A B such that xx A$yy B ( ( x, y ) ) f and ( ( x, y1) f ( ( x, y2 ) ) f y1 = = y2Note: f associates with each x in A one and only one y in B.4Definition 2A is called the domain and B is called the codoma

3、in.If f(x) = y y is called the image of x under f x is called a preimage of y(note there may be more than one preimage of y but there is only one image of x).The range of f is the set of all images of points in A under f. We denote it by f(A). If f is a function from A to B,we say that f maps A to B

4、.5x1x2x3y1y2Yy3XExample 1 f(x1) = y1 the image of x3 is y2 the domain of f is X = x1, x2, x3 the codomain is Y = y1, y2, y3 f(X) = y1, y2 the preimage of y1 is x1 the preimages of y2 are x2 and x3 f(x2, x3 ) = y26Definition 3Let f1 and f2 be function from A to R. Then f1 + f2 and f1 f2 are also func

5、tions from A to R defined by (f1 + f2 )(x) = f1(x) + f2(x), (f1 f2)(x) = f1(x) f2(x).7Definition 4Let f be a function from the set A to the set B and let S be a subset of A.The image of S is the subset of B that consists of the images of the elements of S.We denote the images of S by f(S) , so that

6、f(S) = f(s) | s in S.8x1x2x3y1y2Yy3Xy4One-to-oneInjections(单射）单射）Let f be a function from A to B.Definition 4 f is one-to-one (denoted 1-1) or injective if preimages are unique.Note: this means that if a b then f(a) f(b).9Example 2 Determine whether the function f(x)=x2 from the set of integers to t

7、he set of integers is one-to-one.Solution: The function f(x)=x2 is not one-to-one because, for instance, f(1)=f(-1),but 1-1. Determine whether the function f(x)=x+1 is one-to-one.Solution: The function f(x)=x+1 is a one-to-one function. 10 x1x2x3y1y2XYOnto (到上映射)Surjections（满射）（满射）Let f be a functio

8、n from A to B.Definition 5: f is onto or surjective if every y in B has a preimage.Note: this means that for every y in B there must be an x in A such that f(x) = y.11Example 3 Determine whether the function f(x)=x2 from the set of integers to the set of integers is onto.Solution: The function f(x)=

9、x2 is not onto since there is no integer x with x2= -1 . Determine whether the function f(x)=x+1 from the set of integers to the set of integers is onto. Solution: The function f(x)=x+1 is a onto function. 12x1x2x3y1y2Yy3XOne-to oneand ontox1x2x3y1y2Yy3Neither one-to-one nor ontoXx1x2x3y1y2Yy3Not a

10、functionXBijection（双射）（双射）f is bijective if it is surjective and injective.The function f is a one-to-one correspondence, or a bijection, if it is both one-to-one and onto.13Example 4Let f be the function from a,b,c,d to 1,2,3,4 with f(a)=4, f(b)=2, f(c)=1, and f(d)=3. Is f a bijection?Solution: The

11、 function f is one-to-one and onto. Hence, f is a bijection.14Note:Whenever there is a bijection from A to B, the two sets must have the same number of elements or the same cardinality.15Example 5Let A be a set. The identity function on A is the function A: AA (IA: AA ) A(x)=xwhere xA. The identity

12、function A is the function that assigns each element to itself.The identity function A is one-to-one and onto,so that it is a bijection.16Inverse FunctionsDefinition 6 Let f be a bijection from A to B. Then the inverse of f, denoted f-1, is the function from B to A defined asf-1(y) = x iff f(x) = yA

13、 one-to-one correspondence is called invertible since we can define an inverse of this function. A function is not invertible if it is not a one-to-one correspondence since the inverse of such a function does not exist. 17Example 6 Let R be the set of reals, f:R R, f=(x,x+1)|xR The function f is inv

14、ertible since it is a bijection . The inverse function is : f-1 = (x+1,x)|xR Note: No inverse exists unless f is a bijection.18A B CabcdVWXYhijabcdhijA CgffogNote: The composition fog cannot be defined unless the range of g is a subset of the domain of f.CompositionDefinition7 Let f: BC, g: AB. The

15、composition of f with g, denoted fog, is the function from A to C defined by fog(x) = f(g(x)19Example 7Let X,Y,U be sets of real, f：XU f(x)=1+x2 g: U Y g(u)=SinuWhat is the composition of f and g?Solution gof: XY gof(x)= g(f(x)= Sin(1+x2)20Example 8Suppose f: BC, g: AB and fog is injective. What can

16、 we say about f and g? We know that if a b then f(g(a) f(g(b) since the composition is injective. Since f is a function, it cannot be the case that g(a)= g(b) since then f would have two different images for the same point. Hence, g(a) g(b)It follows that g must be an injection.However, f need not b

17、e an injection .21A B CabcVWXYhijabchijgffogIf there is an element V in B which has an image i in C under f ,but has not a pre-image in A under g , then f(V)=i. The element i must has a pre-image c in A under fog. Since f and g are function, there exits an element X which is the image of c and the p

18、re-image of i. That is f(X)=i. Hence, X Y but f(X)=f(Y). f need not to be an injection.22Example 9If f(x) = x2 and g(x) = 2x + 1, then f(g(x) = (2x+1)2 and g(f(x) = 2x2 + 1Hence, fog and gof are not equal.The commutative law does not hold for the composition of function.23Theorem 1Let f is a one-to-

19、one correspondence between A and B,then (a) f -1of=IA(b) fof -1=IBProofSince f is a one-to-one correspondence from A to B,then the inverse function f 1 exists and is a one-to-one correspondence from B to A. The inverse function reverse the correspondence of the original function, so that f 1(b)=a wh

20、en f(a)=b, and f(a)=b when f 1(b)=a .24Hence,f -1of(a)= f 1(f(a)= f 1(b)=aandf of -1(b)= f (f 1(b)= f (a)=b.Consequently f -1of = IA f of 1 = IB where IA and IB are the identity functions on the sets A and B,respectively.(f 1) 1=f25SOME IMPORTANT FUNCTIONS nThe floor function assigns to the real number x the largest integer that is less than or equal to x. The value of the floor function at x is denoted by x . The ceiling function assigns to the real number x the smallest integer that is greater than or equal to x . The val