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2 SET NOTATION The notation of set is { }. Examples. CHAPTER 1 : FUNCTIONS 1. A is the set of even numbers less than 1. A = {,, 6, 8, 10 }. B = { violet, indigo, blue, green, yellow, orange, red } B is... We can also use Venn Diagram to represent a set. RELATION A relation from set A to set B is the linking (or pairing) of the elements of set A to the elements of set B. 1.1 Represent a relation A relation between two sets can be represented by :- (a) arrow diagram (b) ordered pairs (c) graphs Example. A relation from set A = { -, 1,, ) to set B = { 1,, 9 } is given by power of for. Represent the relation using (a) arrow diagram

3 (b) ordered pairs (c) graph Activity 1 1. Given that set C = {,, 5 } and set D = {, 16, 18, 5 }. Represent the relation multiple of between set C to set D using (a) arrow diagram (b) ordered pairs. Represent the given relation using ordered pairs and graph. (a) child of A B Chandra Aisya Razif Chong Hasnul Lee Siva

4 (b) M has factor N Identify domain, codomain, object, image and range of a relation. Definition Domain - First set Codomain Second set Object Elements in the domain Image Elements in the codomain that are linked to the objects. Range set of images Example 1 Draw an arrow diagram to show a relation power of for from set A = { -, -, -,, } to set B = { - 9, -,, 9 }. Then, identify the (a) image for number (b) image for number (c) object for number (d) object for number 9 (e) domain (f) codomain (g) range of the relation.

5 Solution Example A relation is given in the form of ordered pairs, H = { (, 6), (, 8), (,10), (,8), (5,10), (5,15) }. Identify the (a) image of 5 (b) object of 8 (c) domain (d) range (e) codomain Example The relation below is shown in the graph. Set B Set A Identify the (a) image of (b) object of 15 (c) domain (d) range (e) codomain

6 Activity 1. Draw an arrow diagram to show a relation add to from set P = { 1,,,, 5 } to set Q = { 1,,,, 5, 6, 7, 8 }. From the diagram, state the (a) image of (b) object which its image is 6 (c) domain (d) codomain (e) range for the relation.. The relation that mapped set A = {,,, 5} onto set B = { 6, 8, 9 } is factor of. (a) Represent the relation using (i) arrow diagram (ii) ordered pairs (iii) graph (c) State the (i) domain (iii) range (v) object of (ii) codomain (iv) image of 8 1. Classifying the types of relation Type of relation 1. One to one relation Example of Arrow diagram Each object in the domain has only one image in the codomain. One to many relation Each object in the domain has more than one image in the range

7 . Many to one relation Each image in the range has more than one object in the domain but each object has only one image.. Many to many relation Each object in the domain has more than one image in the range and each image in the range has more than one object in the domain. State the type of relation shown by each of the following relation. Question 1. A element of B Type of relation Metane Ozone Carbon Hydrogen Oxygen. R = { (1,1), (,8), (,7), (,6), (5,5) }

8 . P power of Q B = { (,7), (,9), (,11), (5,9), (6,11) } 5. 5 Set B 1 1 Set A

9 . FUNCTIONS.1 Definition of a functions A functions is a special relation whereby for EVERY object in the domain, there is ONE AND ONLY ONE image in the codomain. Hence, the type of relations that are considered as a functions are (i) one to one relation (ii) many to one relation Examples Identify which of the following relations are functions. Give reason for your answer. Question (a) R = { (1,), (,), (5,6), (7,8) } Answer / Reason (b) A multiple of B (c) M 1 three times N 6 9 (d) K = { (May,1 days),(june,0 days), (July, 1 days),(feb,8 days)}

10 Activity Determine whether each of the following relation are considered as a function or not. Question Function / not a function 1. H = { (1,), (,), (5,7), (8,9) }. A March B 8 days April 0 days May 1 days. A B C D X Y Z. Set B Set A

11 . Express functions using function notation (1) A function can be represented using small letter such as f,g, h and etc. () (a) f : x y ( is read as function f maps x to y ) can also be written as f(x) = y. (b) g : x 5x -, hence g(x) = 5x - ( is read as 5x is the image of x under the function g ) (c) h : m m, hence h(m) = m. (d) Square of 16 f : x x or f(x) = x 6 6 domain codomain. Domain, object, image and range of a function Example 1 x g x Domain = { 0, 1, } Objects =.,.,.. Images =.,.,.. Range = {.,..,..} Example Find the image of f : x and range. x + given that x = { - 1, 1, }. State the domain, object

12 . Determine the image of a function given the object or vice versa. Example 1 (a) Given that f(x) = x 1. Find the image of x =. (b) Given that f ( x) x. Evaluate the value of f ( ). (c) A function h is defined as h : x 5x. Find the image for x = -. (d) Given the function m(x) = x the image for x = Find Example A function g is defined as g : x (a) g(0), g(- ), g(5), g(8) x, find the value of (b) State the value of x such that the function is not defined. Example Given that f : x x + 5. Find the object for which the image is 9.

13 Example Given the function g(x) = x 9. Find the value of the object that maps onto itself. Example 5 Given that function f(x) = x. Find the value of x such that f(x) = x. Activity 5 1. A function f is defined as f : x x +. Find a. f() b. f() c. f(x + 1).. A function f is defined by f : x x. Find (a) the object that has 9 as its image. (b) the object that is mapped onto itself.. A function f is defined as f : x x 1. (a) Find the image of -, -1, 0, 1,. (b) Find the objects that has 9 as it image.

14 . COMPOSITE FUNCTION.1 Composition of two functions f g A B C x f(x) g(x) (1) The figure above shows a function f that maps elements in set A to those in set B and the function g that maps set B to set C. The combined effect of the two functions can be represented by the function gf which is called the composite function of g and f. () fg = fg(x) = f[g(x)] gf () f = ff, f = fff or ff or f f and so on. Example For each pair of the following given function, find the composite function fg and gf. (a) f(x) = 5x, g(x) = x + (b) f(x) = 5x 1, g(x) = x +

15 (c) f : x x, g :x x (d) f : x 5, g : x 1 x x. Determine the image or object of a composite function Examples; 1. Given, f : x 5x and g : x x. Find (a) fg(x) (c) gf(x) (b) fg() (d) gf(- 1). Given that f : x, x 0 x (a) fg(x) and g :x x + 1. Find (b) gf(x)

16 (c) fg() (d) gf(). Given that f : x + x and g : x 8 x, x 0. Find the value of fg().. Functions f and g are defined as f :x x + and g : x x respectively. Find (a) f (b) g (c) f () (d) g ( ) 5. Functions f and g are defined as f(x) = x and g(x) = x 1 respectively. Find the value of x if gf(x) = 11.

17 Activity 6 1. Given f : x x + and g : x x 6. Find (a) fg ( x ) (b) gf ( x ) (c) gf ( ). Functions f and g are defined as f : x x - 5, g : x, x 0 respectively. x (a) Find the composite function gf and the value of gf(). (b) Determine f (x). Given f : x px + q and f : x 5x 8. Find the value of p and q ( p > 0).. INVERSE FUNCTION.1 Find the object, given its image and function by inverse mapping. f x f(x) f - 1 If f represent a function, the inverse function of f is denoted by f 1. ( take note that 1 f 1 f ) Therefore, if f : x y then f(x) = y if f 1 : y x then f 1 (y ) = x Example Function f is defined by f : x x 1.Find the value of a. f 1 (5) b. f 1 (- ) c. f 1 ()?

18 . Determining the inverse function Examples 1. Given f(x) = x + 1. Determine the inverse function f 1.. A function g is defined as g ( x ) = 5x - 8, determine g 1 (x).. Given that f : x x 1, find the inverse function f 1 (x). 5. A function g is defined as g : x x. Determine (a) g 1 (x) (b) the value of g 1 () (c) the value of k if g 1 (k)= k (d) g 1 h(x) if given h : x. x

19 5. Given that f : x x, x and g : x x. Find (a) f 1 (x) and state the value of x such that f 1 is not defined. (b) the value of f 1 g() 6. Given f : x x + k and its inverse function f 1 : x nx +. Find (a) the value of k and n (b) the value of f 1 f() 7. Given f(x) = x + 7 and g(x) = x + 1. Find (a) fg 1 (x) (b) the value of x such that gf(- x) = - 9

20 8. Given f : x p qx. determine (a) f 1 (x) in terms of p and q (b) the value of p and q if f 1 (8) = -1 and f() = - 7. Activity 8 1. Find the inverse function for each of the following (a) f : x x + (b) g : x x 7 (c) f : x, x 0 x (d) g : x x (e) f : x, x x. Given the function f : x x+, find the value of f 1 (5).. A function f is defined as f : x x 5, x 5 x 5. Find the value of f 1 (1) x a. Given that f : x, x x (a) the value of a (b) the value of f 1 (- ) and f(7) =. Find 5. Given the function f : x mx + n and its inverse function value of m and n. f x 8 : x. Find the 1-1

21 . The condition for the existence of an inverse function There are two types of relation that are considered as a functions that is (i) one to one (ii) many to one The reverse mapping for one to one relation is still one to one but the reverse mapping for many to one relation is not a function because it is one to many relation. Example Determine whether each of the following functions has inverse functions. Explain your answer. Question Answer (a) f (b) g.6 5.

22 Topical Test 1. Express the relation factor for between Set A = {, 6, 10, 15} and Set B = {,, 5} in the form of (a) graphs (b) ordered pairs. The arrow diagram shows the relation between Set A and Set B. r. s. t State the (a) domain (b) codomain (c) range (d) image for s (e) object for image f - 1 The figure above shows a function (a) the value of a and b (b) the image for x =. f : x ax bx. Find. Given the function f ( x) x 1 and g ( x) x. Find the composite functions (a) fg (b) gf (c) f ( x ) (d) g ( 1)

23 5. Given f ( x), x and g ( x) x 1. Find the value of x (a) fg (1) (b) g ( ) 6. Sketch the graph for the absolute function f ( x) 5 x in the domain 1 x. Hence, state the range of the function. 7. Given the function f : x x and the function g : x, x 0. Find x (a) the inverse function f 1 ( x) (b) the value of f 1 g() 8. Given the function Find f : x 10, where f ( 1) 5 and f ( ) 10. hx k (a) (b) the value of h and k the values of x such that f(x) = x 10. Functions f and g are defined as f x : x h x, and g : x x 1 (a) State the value of h. (b) Express the functions gf and f 1 ( x) (c) 1 Find the value of f g( ) 11. Given f(x) = px + q, f(0) = -5 and f() = 7. Find the value of p and q