Introductory part of function for class 12th JEE
- 3. FUNCTIONS In day to day life, we come across machines which accept input and give output or outputs. INTRODUCTION When one input is given to a machine it gives one output. It is single task machine And a machine which gives several outputs for a single input is a multi task machine. A function is nothing but a single task machine.
- 4. FUNCTIONS and gives output y, f is single task machine (Function) Here, f is the name of the machine Machine (f) Input(x) Output(y) which takes input x denoted as f(x)=y
- 5. FUNCTIONS g is multi task machine (Not a function) Here, g is the name of machine Input(x) Output(y1) Machine (g) Output(y2) giving outputs y1, y2 for single input x, denoted as g(x)= y1, g(x)=y2
- 6. FUNCTIONS (i.e.) machine f takes input 0 and gives output 3 Let us see one Example (i.e.) machine f takes input -4 and gives output -5 (i.e.) machine f takes input 2 and gives output 7 Consider f(x) = 2x+3(i) (i) If x = 0 (ii) If x = 2 (iii) If x = -4 f(0)=3 f(2)=7 f(-4)=-5 f(-4) = 2(-4)+3 By observing all these three conditions We get different outputs for different inputs Thus, f is such that for single input, only one output exists (i.e.) single task machine. Thus, strongly f is a function. f(0) = 2(0)+3 f(2) = 2(2)+3
- 7. FUNCTIONS Now 2 is prime g(2)=0 Also 2 is even g(2)=1 Thus g(2) = 0 or 1 Example (i.e.) machine g takes single input 2 and gives output 0 or 1 (i.e.) multi task machine. Thus g is not a function Consider g(x)= 0 (x=prime) 1 (x=even) In this case we get two different outputs for single input
- 8. FUNCTIONS 1. A machine f takes input x and gives output y, denoted as 1) x=y 2) f(x)=y 3) f(y)=x 4) none
- 9. FUNCTIONS 2. If f(x)=5x+3, then f is 1) Single task machine 2) Multi task machine 3) Function 4) Both 1 & 3
- 10. FUNCTIONS INTRODUCTION OF DOMAIN & RANGE
- 11. FUNCTIONS Every machine has some limitations in accepting inputs. DOMAIN AND RANGE Domain = set of all acceptable inputs Range = set of all possible outputs. The set of all acceptable inputs of a machine f is called its domain. The set of all resulting outputs is called its range. Thus
- 12. FUNCTIONS Flour mill Input(solid) Output(powder) Input(cloth) No output Flour mill Let us observe inputs and outputs diagrammatically
- 13. FUNCTIONS f is a function whose inputs and outputs are listed in the table: Thus Where 1st Coordinates a, b, c, d ILLUSTRATION Also we write f={(a, 3), (b,7),(c, 8), (d, 0)} f(b) = 7, f(c ) = 8, f(d) = 0 represent inputs taken by f, 3, 7, 8, 0 are corresponding outputs. Input a b c d Output 3 7 8 0 f(a) = 3,
- 14. FUNCTIONS a b c d 3 7 8 0 f Here f establishes connection between set of inputs {a, b, c, d} to set of outputs{3, 7, 8, 0}. Thus range = {3, 7, 8, 0} Let us observe the previous table diagrammatically domain = {a, b, c, d}, Do you know domain & range?
- 15. FUNCTIONS Let us know the preimage and image of a function The terminology input and output can be viewed as pre-image & image under f as follows f(a)=3 f(b)=7 f takes input ‘a’ a b c d 3 7 8 0 f and gives output 3 3 is image of ‘a’ under f f takes input ‘b’ and gives output 7 Pre-image of 7 is ‘b’ under f
- 16. FUNCTIONS 1. Domain is nothing but 1) Set of all possible outputs 2) Set of all acceptable inputs 3) Inputs 4) none
- 17. FUNCTIONS 2. If f={(p,1), (q,2), (r,3)} then domain of f is 1) {1,2,3} 2) {p,q,r} 3) {p,q,r,1,2,3} 4) none
- 18. FUNCTIONS 3. If f={(p,1), (q,2), (r,3)} then which of the following is true 1) 2 is the image of q 2) r is the preimage of 3 3) {1,2,3} is the range of f 4) All the above
- 20. FUNCTIONS MATHEMATICAL DEFINITION OF FUNCTION A, B are two non-empty sets. A rule f is such that “f connects every element in A to a unique element in B”. Then, f is called as a function from A to B and is written as f: A B (or) A 𝒇 𝑩 (read as f maps A to B). Observe that an element in A should not be mapped to two or more elements in B. Also A, B are called as domain of f, co-domain of f respectively This is an important point we have to remember
- 21. FUNCTIONS f maps aA to bB is written as f(a)=b. Set of all images is range and range is sub set of co- domain. Usually functions are named as f, g, h, ……… Note We use the terms “function”, “map”, “mapping”, “correspondence” as interchangeable. Also f(a)=b means ‘b’ is image of ‘a’ under f or pre-image of ‘b’ is ‘a’.
- 22. FUNCTIONS a b c d 1 2 3 4 f B A EXAMPLE Is ‘f ’ a function or not? Here ‘c’ has no image in ‘B’ f is not a function because c in A has no image in B NO Why?
- 23. FUNCTIONS x y z p q r s g B A EXAMPLE Is ‘g’ a function or not? NO g is not a function because element y in A has 2 connections q, s in B. Here, ‘y’ has two images in ‘B’ Why?
- 24. FUNCTIONS DOMAIN CO DOMAIN f is a function and domain of f = {a, b, c}, co-domain of f = {x, y, z} a b c x y z f EXAMPLE Is ‘f’ function or not? yes Range of f = only Images = {x, y}. Then what is the range of f ? Thus, range is subset of co-domain.
- 25. FUNCTIONS 1 2 3 4 a b c d h EXAMPLE DOMAIN CO DOMAIN Is ‘h’ function or not? yes Then what is the range of h? h is a function Domain of h={1, 2, 3, 4}, Co-Domain of h={a, b, c, d} Range of h={a, b, c, d}. Thus, in this example, range = co-domain.
- 26. FUNCTIONS then {(a, b)/aA, bB} and f(a)=b plotted in XY plane gives graph of f. GRAPH OF A FUNCTION Let f : A B be a function such that f(a) = b for a A, bB, Note If the graph of f(x)=y plotted in XY plane is such that a line parallel to y-axis cuts the graph in more than one point, Then f(x)=y is not a function.
- 27. FUNCTIONS y x O Line parallel to y-axis y=f(x) y=f(x) shown in above graph is not a function. Let us observe this graph
- 29. FUNCTIONS Set If A, B are two non–empty sets then a set {(a,b)/aA, bB} is called Cartesian product of A & B AB={(a,b)/aA, bB} Well defined collection of objects is called a set. Cartesian product Example A = {1, 2, 3, 4} It is denoted by AB Before going to the relation we have to know set and Cartesian product
- 30. FUNCTIONS A = {1, 2, 3}, B = {a, b} AB= (2,a), (2,b), (3,a), (3,b)} Ex: Relation If f ⊆ A×B then f is called a relation from A to B. {(1,a), (1,b),
- 31. FUNCTIONS DEFINITION OF FUNCTION THROUGH RELATION For all a A, ordered pair (a , b) f for some b B and (a, b) f, (a, c) f b = c (i.e.,) ordered pairs with same first co-ordinate and distinct second co-ordinates are not allowed. Then f is said to be function from A to B If f A B is a relation satisfying
- 32. FUNCTIONS CO DOMAIN DOMAIN f = {(1,1), (2, 3), (3, 1), (4, 4)} is a function from {1, 2, 3, 4} to itself. 1 2 3 4 1 2 3 4 f A A Example (1)
- 33. FUNCTIONS as occurs same first co-ordinate in two ordered pairs. 1 2 2 3 g A B g = {(1, 2), (1, 3), (2, 3)} is not a function Example (2)
- 34. FUNCTIONS Sets A, B contains m, n elements respectively i.e., No. of relations from Set A to Set B No. of functions from Set A to Set B Important Rules n(A)=m, n(B)=n =2mn = 2n(A)n(B) = nm = n(B)n(A)
- 35. FUNCTIONS Ex (1) No. of relations from Set of 3 elements to set of 4 elements = 2(3)(4) No. of functions from Set of 3 elements to set of 4 elements Ex (2) Can you guess the answer? = 43 Can you guess the answer? = 212
- 36. FUNCTIONS EQUALITY OF FUNCTIONS Two functions f and g are said to be equal if (i) f and g are defined on the same domain (ii) f (x) = g (x) for all x domain of f Example 1 Let f:RR, g:RR, be defined as f(x)=x2 for every x R and g(y)=y2 for every y R; Then the functions f, g are equal (i.e) f=g Here, x, y are simply dummy variables.
- 37. FUNCTIONS Consider f (x) = log x2, g (x) = 2 log x Thus the domain of f domain of g f (x) is defined every where except at x=0 g (x) is defined when x > 0 Example 2 f g That is domain of f(x) is R-{0} That is domain of g(x) is R+ By observing both the domains
- 38. FUNCTIONS On what domain the function f (x) = x2 - 2x, g (x) = -x + 6 are equal f (x) and g (x) are equal on the domain {-2, 3} f (x) = g (x) (x + 2) (x - 3) = 0 Example 3 x2 -2x = -x + 6 x2 – x -6 = 0 x = -2, 3
- 39. FUNCTIONS 1. A={p,q,r}, B={1,2}, then AB is 1) AB={(p,1),(q,2),(r,2)} 2) AB={(p,1),(p,2),(q,1),(q,2)} 3) AB={(p,1),(p,2),(q,1),(q,2),(r,1),(r,2)} 4) none
- 40. FUNCTIONS 2. n(A)=2, n(B)=4, then number of relations from setA to setB 1) 22 2) 24 3) 224 4) 42
- 41. FUNCTIONS 3. n(A)=2, n(B)=4, then number of functions from setA to setB 1) 22 2) 24 3) 224 4) 42