5. Ways of Expressing a Relation 4. Graph 1. Set notation 5. Mapping 2. Tabular form 3. Equation
6. Example: Express the relation y = 2x;x= 0,1,2,3 in 5 ways. . 1. Set notation (a) S = { ( 0, 0) , ( 1, 2 ) , ( 2, 4 ), ( 3, 6) } or (b) S = { (x , y) such that y = 2x, x = 0, 1, 2, 3 } 2. Tabular form
7. y 5 5 4 3 2 1 x 5 -4 -2 1 3 5 -5 -1 4 -3 2 -5 -1 -2 -3 -4 -5 -5 3. Equation: y = 2x 5. Mapping 4. Graph x y ● 0 0 ● 1 2 ● 2 4 6 3
8. DEFINITION: Domain and Range All the possible values of x is called the domain and all the possible values of y is called the range. In a set of ordered pairs, the set of first elements and second elements of ordered pairs is the domain and range, respectively. Example: Identify the domain and range of the following relations. 1.) S = { ( 4, 7 ),( 5, 8 ),( 6, 9 ),( 7, 10 ),( 8, 11 ) } Answer : D: { 4,5,6,7,8} R:{7,8,9,10,11}
9. 2.) S = { ( x , y ) s. t. y = | x | ; x R } Answer: D: all real nos.R: all real nos. > 0 3) y = x 2 – 5 Answer. D: all real nos. R: all real nos. > -5 4) | y | = x Answer: D: all real nos. > 0 R: all real nos.
10. g) Answer: D: all real nos. except -2 R: all real nos. except 2 5. Answer : D: all real nos. > –1 R: all real nos. > 0 6. Answer: D: all real nos. < 3 R: all real nos. except 0 7.
11. Exercises: Identify the domain and range of the following relations. 1. {(x,y) | y = x 2 – 4 } 2. 5. 7. y = 25 – x 2 y = | x – 7 | 6. 4. 3. 8. y = (x 2 – 3) 2 9. 10.
12. PROBLEM SET #5-1 FUNCTIONS Identify the domain and range of the following relations.
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15. is an example of a many-to-one function is an example of a one-to-one function Functions One-to-one and many-to-one functions Consider the following graphs and One-to-many is NOT a function. It is just a relation. Thus a function is a relation but a relation could never be a function.
16. Example: Identify which of the following relations are functions. a) S = { ( 4, 7 ), ( 5, 8 ), ( 6, 9 ), ( 7, 10 ), ( 8, 11 ) } b) S = { ( x , y ) s. t. y = | x | ; x R } c) y = x 2 – 5 d) | y | = x e) f)
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18. When we use f as a function, then for each x in the domain of f , f ( x ) denotes the image of x under f .
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20. Piecewise Defined Function A piecewise defined function is defined by different formulas on different parts of its domain. Example: if x<0 if
21. Piecewise Defined Function EXAMPLE: Evaluate the piecewise function at the indicated values. if x<0 f(-2), f(-1), f(0), f(1), f(2) if if if if f(-5), f(0), f(1), f(5)
22. DEFINITION: Operations on Functions If f (x) and g (x) are two functions, then Sum and Difference ( f + g ) ( x ) = f(x) + g(x) Product ( f g ) ( x ) = [ f(x) ] [ g(x) ] Quotient ( f / g ) ( x ) = f(x) / g(x) d) Composite ( f ◦ g ) ( x ) = f (g(x))
23. Example :1. Given f(x) = 11– x and g(x) = x 2 +2x –10 evaluate each ofthe following functions f(-5) g(2) (f g)(5) (f - g)(4) f(7)+g(x) g(-1) – f(-4) (f ○ g)(x) (g ○ f)(x) (g ○ f)(2) (f○ g)
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27. We can also find the domain and range from the graph of a function.
29. Graph of piecewise defined function The graph of a piecewise function consists of separate functions. Example: Graph each piecewise function. if if if if if
32. Graph of absolute value function. Recall that if if Using the same method that we used in graphing piecewise function, we note that the graph of f coincides with the line y=x to the right of the y axis and coincides with the line y= -x the left of the y-axis.
33. Example: Graph each of the follow functions. y = | x – 7 | y = x-| x - 2 | 1. 4.
36. Definition: Greatest integer function. The greatest integer function is defined by greatest integer less than or equal to x Example: 1 3 0 -4 1 0 -1 1 0 2 0 2 1
37. Definition: Least integer function. The least integer function is defined by least integer greater than or equal to x Example: 2 4 0 -3 2 1 0 2 1 2 1 3 1
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