University:
University of Michigan
Course:
ECON 613 | Advanced Topic in Mathematics for Economists
Academic year:

2022
Views:

154

Pages:

6
Author:

NebulaNavigator

Types of Functions 1. One-One function or One-to-one function or Injective or Injection or A functoon f: A→B is said to be one-one if of maps distinct elements of A to distinct elements of B se f is one-one if x 10 = few) ≠ few OR fis one-one if ofer) = fon) x1=12 A fi B 1 12 3- 17 5 19 25 f, is one-one AB 1- 312 3 17 5- 25 f2 is NOT one-one Here 13, but their imaged. are the same 2 Onto function or Surjective or Surjection A function f: A B is sard to be onto if range of f = B ie if range of f = Co domain of f. ie f: A B is onto if for every yEB, there exists x E A Such that fey=y es. A B 2. 25 9 34 10 45 IHere range of f = B - fi is Onto fa B A 2- 21 6 534 9 45 10 Hese range = {25, 34-45] range B • fe is NOT onto 3 Bijective function or Bijection A function f: A B is said to be a bijection if f is both one-one and onto. If F:A→B is a bijection, then we say there is a one-to-one correspondence between. A and B Nolet Consider the following functions. A fi B a 5 C. 7 d 8 f, is one-one fi is not onto. Ge fi maps different elements to different elements) EG 7 V 10 3 P12 2 fa is one-one fr is Onto. ie fa is one-one and onto fa is a bijectim. S fa H a b C 33 4 5 e fo is not one-one f3 is Onto Inverse of a function Let f: AB be a function. The inverse of f, denoted by f-1, is a function form B tolt Such that fen=y) f(y) = es: f(2)=6> f-1(6)=2 Nolit f: A→B and f: BA Result The inverse of a function exists if and only if the function is one-one and onto. 6 inverse exists if and only if the functimis a bijestion. Defnt A function of is said to be invertible if f' exists. Nolit A function of is invertible if and only if f is one-one and onto. I Find the domain and inverse of the function fou = x²+5 fou is defined for all read Valwes of Domain of f, D(f) = R To find the inverse Let few = -f() Now, f = 13+5 y = 13+5 2 Find the domain and inverse of the function fa) = DL-2 1-3 Pul Dro 6x-3=0 21:3 feu is not defined for 2 =3 [ie when 21.3 the Dr of feu =] - Domain = R-{3} To find the inverse Let few y Then L = f(y). Now, f) = 21-2 31-3 y = 21-2 y (1-3) = ху-зу 21-2 2+39 (-1) 39-2 y-1 f(y) = 37-2 f(x) = 31-2 3 Find the domain of fer) = 7x fey = 71 Puk Dr = 0 (21-2)(1-3) 20 31-2-0 ur x-3=0 ) X = 2 x=3 x = 2,3 for is not defined for 1 = 2 and 1=3 Domain R-{2,3} 4 Find the domain of fou= x2-4 f(x) = x²-4 Put Dr20 10 - ferd is not defined for 1=0 Domain = R-{0}